#6481952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 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)(9,10)(11,16)(12,15)(13,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(1,3,4,8)(2,5,6,7)(9,11,12,13)(10,14,15,16)(17,19)(18,22)(20,27)(21,25)(23,31)(24,29)(26,28)(30,32);(10,14)(11,13)(15,16)(19,25)(22,29)(26,27)(30,31);(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31);(10,15)(14,16)(18,24)(22,29)(23,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389148);