#9481449

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3898 conjugacy classes of subgroups; only the 2850 normal subgroups are shown.

normal subgroups only

This group has 3898 conjugacy classes of subgroups — too many to draw. Only its 2850 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 72 points:

(1,2)(3,7)(4,5)(6,8)(9,10,15,20)(11,16,21,27)(12,23,25,17)(13,18,26,33)(14,19,30,36)(22,31,29,35)(24,37,39,32)(28,34,40,38)(41,42)(43,49)(44,45)(46,50)(47,61)(48,52)(51,62)(53,63)(54,58)(55,57)(56,64)(59,70)(60,66)(65,71)(67,69)(68,72);(2,5)(6,7)(11,21)(13,26)(16,27)(18,33)(22,29)(28,40)(31,35)(34,38)(41,43,48,44)(42,49,52,45)(46,55,60,53)(47,54,62,56)(50,57,66,63)(51,64,61,58)(59,68,71,67)(65,69,70,72);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,18)(13,17)(14,29)(15,21)(19,35)(20,27)(22,30)(23,26)(24,38)(25,33)(28,37)(31,36)(32,40)(34,39)(41,44,48,43)(42,49,52,45)(46,55,60,53)(47,56,62,54)(50,63,66,57)(51,64,61,58)(59,68,71,67)(65,72,70,69);(1,3)(2,6)(4,8)(5,7)(9,12,15,25)(10,17,20,23)(11,18,21,33)(13,27,26,16)(14,24,30,39)(19,32,36,37)(22,34,29,38)(28,35,40,31)(41,45,48,49)(42,44,52,43)(46,57,60,63)(47,58,62,64)(50,55,66,53)(51,56,61,54)(59,69,71,72)(65,68,70,67);(9,13)(10,18)(11,17)(12,16)(14,28)(15,26)(19,34)(20,33)(21,23)(22,32)(24,31)(25,27)(29,37)(30,40)(35,39)(36,38)(41,46)(42,50)(43,53)(44,55)(45,57)(47,59)(48,60)(49,63)(51,65)(52,66)(54,67)(56,68)(58,69)(61,70)(62,71)(64,72);(9,14)(10,19)(11,22)(12,24)(13,28)(15,30)(16,31)(17,32)(18,34)(20,36)(21,29)(23,37)(25,39)(26,40)(27,35)(33,38)(41,47)(42,51)(43,54)(44,56)(45,58)(46,59)(48,62)(49,64)(50,65)(52,61)(53,67)(55,68)(57,69)(60,71)(63,72)(66,70)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268337, #72268340
Fitting subgroup of	#72268337
maximal subgroup of	#72268306, #72268337
maximal normal subgroup of	#72268306, #72268337
maximal quotient of	#72268312

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388645);