#134862

order	512
nilpotency class	4
isoclinism	isoclinic class of #134769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	#427
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

231 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 90 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)(18,24)(19,30)(20,25)(21,22)(26,42)(27,39)(28,38)(29,40)(31,46)(32,48)(33,36)(34,35)(37,44)(45,47);(17,18,25,39,22,28,35,40,23,29,34,38,21,27,20,24)(19,31,41,46,30,45,48,37,33,26,43,42,36,44,32,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,33)(18,26,29,31)(20,32,35,48)(21,36,22,30)(24,37,40,47)(25,41,34,43)(27,45,28,44)(38,46,39,42)

Polycyclic presentation:

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

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, 42058);