#134685

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₄⋊C₈
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cgi

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, 41881);