#379521

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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

1172 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fg,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg

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, 286717);