#354540

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2)(3,4)(6,8)(9,10,12,11)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(2,4)(13,15,16,22,17,21,23,27)(14,18,20,25,19,24,26,28);(5,6,7,8)(15,21)(18,24)(22,27)(25,28);(9,11,12,10)(14,19)(15,21)(20,26)(22,27)

Polycyclic presentation:

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

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, 261736);