#7506936

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	#31728	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1505 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)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(23,28)(26,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(18,23)(21,27)(22,28)(24,30)(26,31)(29,32);(10,14)(13,16)(17,18,20,22)(19,24,25,29)(21,28,27,23)(26,32,31,30);(2,7)(3,8)(10,14)(13,16)(17,19)(18,24)(20,25)(21,26)(22,29)(23,30)(27,31)(28,32);(1,4,5,6)(2,3,7,8)(19,25)(24,29)(26,31)(30,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,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, 7414132);