#844912

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(5,6)(7,8)(9,10,13,15)(11,12,16,14)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(9,11,13,16)(10,14,15,12)(17,18)(19,22)(20,21)(23,24);(2,4)(11,16)(12,14)(18,21)(22,24);(6,8)(9,12,13,14)(10,11,15,16)(18,21)(22,24);(17,19,20,23)(18,22,21,24)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df

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, 752108);