#144199

order	512
nilpotency class	4
isoclinism	isoclinic class of #144119
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	C₂³	#496
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

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

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, 51395);