#382095

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

984 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=efh,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, 289291);