#6435398

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#62686	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3034 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 3034 conjugacy classes of subgroups — too many to draw. Only its 610 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 — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 6342594);