#356791

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=eg,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263987);