#7234149

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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

1590 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)(9,10)(11,16)(12,15)(13,14)(18,22)(19,26)(20,24)(23,28)(25,32)(29,30);(2,5)(3,7)(10,14)(11,12)(18,23)(19,26)(22,28)(27,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,18,24,28,21,23,20,22)(19,25,31,32,26,30,27,29);(1,3,4,7)(2,6,5,8)(10,14)(15,16)(17,19,21,26)(18,25,23,30)(20,27,24,31)(22,29,28,32);(2,5)(6,8)(9,12,13,11)(10,16,14,15)(19,26)(25,30)(27,31)(29,32)

Polycyclic presentation:

2^9|c^2=fgh,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7141345);