#346493

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62871	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2300 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2300 conjugacy classes of subgroups — too many to draw. Only its 188 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209847, #72209849
maximal normal subgroup of	#72209847
maximal quotient of	#72227670, #72227712, #72227718, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253689);