#7605676

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1319 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)(3,7)(4,8)(5,6)(9,10,13,12)(11,14,16,15)(17,18)(19,26)(20,27)(21,24)(22,25)(23,28)(29,30)(31,32);(2,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,26)(20,27)(21,24)(22,25)(23,28)(29,30)(31,32);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,15,16,14)(17,19,26,18,22,24,21,25)(20,27,32,23,29,30,28,31);(1,3,5,4)(2,7,6,8)(11,16)(14,15)(17,20,21,28,22,29,26,32)(18,23,19,27,25,31,24,30);(1,4,5,3)(2,7,6,8)(17,21,22,26)(18,24,25,19)(20,28,29,32)(23,30,31,27)

Polycyclic presentation:

2^9|a^2=ghi,c^2=fhi,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=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, 7512872);