#125967

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

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

normal subgroups only

This group has 310 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

(5,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18)(25,26,31,35)(27,32,30,45)(28,38,40,55)(29,41,44,54)(33,48,50,39)(34,37,52,43)(36,53,46,51)(42,47,56,49);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17)(25,27,31,30)(26,32,35,45)(28,36,40,46)(29,37,44,43)(33,47,50,49)(34,41,52,54)(38,53,55,51)(39,42,48,56);(1,3)(2,5)(4,7)(6,8)(25,28,31,40)(26,33,35,50)(27,36,30,46)(29,42,44,56)(32,47,45,49)(34,51,52,53)(37,48,43,39)(38,54,55,41)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,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, 33163);