#125918

order	512
nilpotency class	4
isoclinism	isoclinic class of #125636
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#31192	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1954 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 376 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:

(1,2,4,7)(3,5,8,6)(13,16)(17,18)(19,30)(20,29)(21,25)(22,28)(23,26)(24,31)(27,37)(32,45)(33,44)(34,41)(35,43)(36,40)(38,42)(39,48)(46,47);(2,5)(3,8)(6,7)(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=fi,b^2=i,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, 33114);