#6966523

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6873719);