#135861

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂³	C₂×C₄⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

257 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,22,19,21,26,31,29,23,25,32,28,20,24,30,27);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19,22,26,21,29,31,25,23,28,32,24,20,27,30,18);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20,23,21)(18,24,25,26)(19,27,28,29)(22,30,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43057);