#99134

order	512
nilpotency class	3
isoclinism	isoclinic class of #99134
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 616 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)(9,10)(11,18,20,14)(12,17)(13,16)(15,19,22,24)(21,23)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(9,11)(10,14)(12,19)(13,20)(15,17)(16,18)(21,24)(22,23)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,16,22)(12,19)(13,20)(14,17,18,23)(21,24)(27,33)(30,36)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,f^b=fg,f^c=fgh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6330);