#353839

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂³
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261035);