#359360

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

1020 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=eh,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di

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, 266556);