#514724

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421920);