#134856

order	512
nilpotency class	4
isoclinism	isoclinic class of #134765
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#107
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

234 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,21,26,23,27,22,37)(19,28,32,44,30,42,33,47)(20,34,35,41,36,48,29,25)(24,38,40,46,39,45,43,31);(17,19,29,45,22,33,36,46,23,30,35,38,21,32,20,31)(18,24,34,44,26,40,41,42,27,39,48,47,37,43,25,28);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(19,30)(24,39)(28,42)(31,46)(32,33)(38,45)(40,43)(44,47)

Polycyclic presentation:

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

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, 42052);