#522096

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

2830 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fg,d^2=g,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, 429292);