#6471745

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(18,24)(19,27)(21,23)(25,32)(28,29)(30,31);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,28)(20,21)(22,24)(23,26)(25,30)(27,32)(29,31);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19)(18,25)(20,29)(21,27)(22,30)(23,31)(24,28)(26,32);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,20,22,26)(18,21,24,23)(19,29,30,32)(25,27,28,31);(10,14)(13,16)(17,21,22,23)(18,26,24,20)(19,27,30,31)(25,32,28,29)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,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, 6378941);