#6966551

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

1419 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)(17,18,22,26,23,28,21,27)(19,24,29,31,25,20,30,32);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,31)(27,32);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,24)(18,25,28,19)(21,32,22,31)(26,29,27,30);(2,6)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,28,27)(19,29,25,30)(20,31,24,32);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,27,28,26)(19,29,25,30)(20,32,24,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,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, 6873747);