#6431838

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×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4024 conjugacy classes of subgroups; only the 906 normal subgroups are shown.

normal subgroups only

This group has 4024 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)(11,15)(12,16)(13,17)(14,18)(19,28)(20,26)(21,27)(22,29)(23,30)(24,31)(25,33)(32,38)(34,37)(35,39)(36,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,28)(17,26)(18,27)(22,32)(23,33)(24,34)(25,30)(29,38)(31,37)(35,40)(36,39);(9,12,14,23)(10,16,18,30)(11,19,21,33)(13,22,24,35)(15,25,27,28)(17,29,31,39)(20,32,34,40)(26,36,37,38);(9,13)(10,17)(11,20)(12,22)(14,24)(15,26)(16,29)(18,31)(19,32)(21,34)(23,35)(25,36)(27,37)(28,38)(30,39)(33,40);(13,24)(17,31)(20,34)(22,35)(26,37)(29,39)(32,40)(36,38)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^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, 6339034);