#495511

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
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

820 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=f,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402707);