#7509536

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30581	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1389 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,25)(19,26,30,24)(20,21,32,23)(27,28,29,31);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,31)(21,29)(22,30)(23,27)(25,26)(28,32);(9,12,13,11)(10,14,15,16)(18,25)(19,27)(21,23)(24,28)(26,31)(29,30);(2,7)(3,8)(10,15)(14,16)(17,20)(18,21)(19,28)(22,32)(23,25)(24,27)(26,29)(30,31);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,20,25,32)(19,29,30,27)(24,31,26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7416732);