#381400

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

1022 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)(6,9)(10,12)(14,17)(18,19)(21,22,25,26,24,28,23,27);(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,19,20,18)(21,23,24,25)(22,26,28,27);(1,2,3,4)(7,11)(10,12)(13,15)(14,18)(16,20)(17,19);(5,7,8,11)(6,10,9,12)(15,20)(18,19)(21,23,24,25)(22,27,28,26)

Polycyclic presentation:

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

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, 288596);