#7492890

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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)(3,6)(4,5)(7,8)(17,18,22,25)(19,26,23,20)(21,24,31,32)(27,30,29,28);(2,5)(3,7)(10,14)(11,12)(17,19,22,23)(18,20,25,26)(21,29,31,27)(24,30,32,28);(9,10)(11,16)(12,15)(13,14)(17,20,23,18,22,26,19,25)(21,30,29,32,31,28,27,24);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21,22,31)(18,24,25,32)(19,27,23,29)(20,28,26,30);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,19,22,23)(18,20,25,26)(21,29,31,27)(24,30,32,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7400086);