#496601

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60682	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

774 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403797);