#144070

order	512
nilpotency class	4
isoclinism	isoclinic class of #143959
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

373 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51266);