#638217

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3718 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 545413);