#7506927

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	#31042	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 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)(17,18,21,22)(19,23,26,29)(20,27,28,24)(25,31,32,30);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(18,22)(19,23)(20,28)(25,31)(26,29)(30,32);(10,14)(13,16)(17,19)(18,23)(20,25)(21,26)(22,29)(24,30)(27,31)(28,32);(2,7)(3,8)(10,14)(13,16)(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31);(1,4,5,6)(2,3,7,8)(20,28)(24,27)(25,32)(30,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=i,e^2=fi,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, 7414123);