#416488

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fh,c^2=e,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi,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, 323684);