#403351

order	512
nilpotency class	3
isoclinism	isoclinic class of #393921
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2249 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 2249 conjugacy classes of subgroups — too many to draw. Only its 282 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310547);