#99129

order	512
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

512 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6325);