#7326983

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=i,e^2=i,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=dh,e^c=ei,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, 7234179);