#134745

order	512
nilpotency class	4
isoclinism	isoclinic class of #134745
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1807
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

306 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fi,e^c=ei,f^c=fhi

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, 41941);