#6471279

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1736 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,24,20,30)(21,28,22,27)(25,29,32,31);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,23)(21,31)(22,29)(25,28)(26,30)(27,32);(11,15)(13,16)(17,20,23,19)(18,25,26,32)(21,31,22,29)(24,27,30,28);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(18,26)(24,30)(25,32)(27,28);(10,14)(13,16)(17,21,23,22)(18,27,26,28)(19,29,20,31)(24,25,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6378475);