#6431840

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431795
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3900 conjugacy classes of subgroups; only the 906 normal subgroups are shown.

normal subgroups only

This group has 3900 conjugacy classes of subgroups — too many to draw. Only its 906 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 — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,g^b=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339036);