#6471895

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1815 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

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

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, 6379091);