#143792

order	512
nilpotency class	4
isoclinism	isoclinic class of #143719
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

352 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50988);