#125966

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

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

normal subgroups only

This group has 334 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) = 256

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

Polycyclic presentation:

2^9|a^2=fi,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, 33162);