#514237

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5498 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=h,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, 421433);