#414937

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efh,c^2=i,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,g^b=gi,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, 322133);