#7492859

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324436
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

2051 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)(19,25)(21,27)(23,30)(24,29)(26,31)(28,32);(2,5)(3,7)(10,14)(11,12)(17,18)(19,26)(20,22)(21,24)(23,28)(25,31)(27,29)(30,32);(9,10)(11,16)(12,15)(13,14)(17,19,27,32,22,28,21,25)(18,23,29,31,20,26,24,30);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,23,28,26)(21,29,27,24)(25,30,32,31);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(18,20)(23,26)(24,29)(30,31)

Polycyclic presentation:

2^9|c^2=gi,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=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7400055);