#522097

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

3174 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)(11,15)(12,13)(14,16)(18,21)(19,20);(2,4)(5,7,8,6)(10,13)(14,15)(17,18,22,21)(19,24,20,23);(1,2)(3,4)(17,19,22,20)(18,23,21,24);(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,24,21,23);(9,11,12,16)(10,14,13,15)(17,21,22,18)(19,24,20,23)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=g,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, 429293);