#125664

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	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
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

374 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,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, 32860);