#7492960

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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

1875 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)(17,18,23,26)(19,28,21,25)(20,22,27,24)(29,30,31,32);(2,5)(3,7)(10,14)(11,12)(17,19)(18,25)(20,29)(21,23)(22,32)(24,30)(26,28)(27,31);(9,10)(11,16)(12,15)(13,14)(17,20,24,18,23,27,22,26)(19,29,30,25,21,31,32,28);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21,23,19)(18,25,26,28)(20,29,27,31)(22,30,24,32);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,22,23,24)(18,20,26,27)(19,30,21,32)(25,31,28,29)

Polycyclic presentation:

2^9|a^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=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7400156);