#6471127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1754 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)(11,13)(12,14)(15,16)(18,22)(19,23)(24,28)(25,27)(26,30)(31,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31);(11,15)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(10,14)(13,16)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)

Polycyclic presentation:

2^9|b^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6378323);