#613015

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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

4914 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 520211);