#424914

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1448 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332110);