#414660

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321856);