#7507839

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=gi,d^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,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, 7415035);