#495472

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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)(9,10,14,20,16,21,15,36)(11,22,25,48,26,49,37,45)(12,18,29,44,27,42,30,59)(13,31,33,46,35,61,34,19)(17,32,39,53,41,62,40,24)(23,47,52,69,50,68,60,67)(28,55,57,66,54,71,58,43)(38,56,64,70,63,72,65,51);(9,11,14,25,16,26,15,37)(10,17,20,39,21,41,36,40)(12,23,29,52,27,50,30,60)(13,32,33,53,35,62,34,24)(18,38,44,64,42,63,59,65)(19,45,31,22,46,48,61,49)(28,56,57,70,54,72,58,51)(43,67,55,47,66,69,71,68);(2,5)(6,8)(9,12)(10,18)(11,23)(13,28)(14,29)(15,30)(16,27)(17,38)(19,43)(20,44)(21,42)(22,47)(24,51)(25,52)(26,50)(31,55)(32,56)(33,57)(34,58)(35,54)(36,59)(37,60)(39,64)(40,65)(41,63)(45,67)(46,66)(48,69)(49,68)(53,70)(61,71)(62,72);(1,3,4,7)(2,6,5,8)(12,27)(18,42)(23,50)(28,54)(29,30)(38,63)(43,66)(44,59)(47,68)(51,70)(52,60)(55,71)(56,72)(57,58)(64,65)(67,69)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,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, 402668);