#135686

order	512
nilpotency class	4
isoclinism	isoclinic class of #135685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1834
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=d,d^2=h,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^b=fgh,f^c=fgh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42882);