#6370907

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4402 conjugacy classes of subgroups; only the 1298 normal subgroups are shown.

normal subgroups only

This group has 4402 conjugacy classes of subgroups — too many to draw. Only its 1298 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,23,26,17)(13,18)(14,19)(15,20)(21,32,25,29)(22,30)(24,36,37,31)(27,33)(28,34)(35,40,38,39);(2,5)(6,8)(9,11,15,14)(10,16,20,19)(12,21,26,25)(13,22,27,28)(17,29,23,32)(18,30,33,34)(24,35,37,38)(31,39,36,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,24)(14,25)(15,26)(16,29)(18,31)(19,32)(20,23)(22,35)(27,37)(28,38)(30,39)(33,36)(34,40);(9,13,15,27)(10,18,20,33)(11,22,14,28)(12,24,26,37)(16,30,19,34)(17,31,23,36)(21,35,25,38)(29,39,32,40);(9,14,15,11)(10,19,20,16)(12,25,26,21)(13,22,27,28)(17,32,23,29)(18,30,33,34)(24,35,37,38)(31,39,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278103);