#125984

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

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

normal subgroups only

This group has 308 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) = 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)(27,38)(28,37)(29,33)(30,36)(31,34)(32,39)(35,45)(40,53)(41,52)(42,49)(43,51)(44,48)(46,50)(47,56)(54,55);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17)(25,27,30,41)(26,32,36,49)(28,42,31,39)(29,44,46,40)(33,47,50,54)(34,52,37,38)(35,53,51,48)(43,55,45,56);(1,3,4,7)(2,5,6,8)(26,33)(28,31)(29,45)(32,47)(34,51)(35,37)(36,50)(38,53)(39,42)(40,55)(43,46)(44,56)(48,52)(49,54)

Polycyclic presentation:

2^9|a^2=fi,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, 33180);