#94858

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄³	C₂³
commutator, socle, prime radical	C₂³	C₄³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,b^a=bi,c^a=ch,c^b=cg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2054);