#522868

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2524 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)(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,19)(20,23,24,22);(9,10,12,14)(11,13,16,15)(18,19)(22,23);(1,2)(3,4)(5,7,8,6)(17,18,21,19)(20,22,24,23);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,23,24,22);(9,11,12,16)(10,13,14,15)(17,20,21,24)(18,22,19,23)

Polycyclic presentation:

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

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, 430064);