#125754

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

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 ◁ #1939 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

347 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,d^2=fi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,h^a=hi,d^b=df,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,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, 32950);