#7506932

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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	#31032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,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, 7414128);