#125799

order	512
nilpotency class	4
isoclinism	isoclinic class of #125636
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

332 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 512

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

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

Polycyclic presentation:

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

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, 32995);