#387328

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1150 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fg,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,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, 294524);