#125735

order	512
nilpotency class	4
isoclinism	isoclinic class of #125647
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1619
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32931);