#6434730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432953
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2808 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 2808 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,23,21)(19,24,30,28)(20,25,33,31)(22,34,35,26)(27,39,41,36)(29,42,43,37)(32,44,45,38)(40,46,48,47);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,36)(26,37)(31,39)(32,40)(33,41)(34,42)(35,43)(38,46)(44,47)(45,48);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,25)(19,27)(21,31)(22,32)(23,33)(24,36)(26,38)(28,39)(29,40)(30,41)(34,44)(35,45)(37,46)(42,47)(43,48);(2,8)(6,14)(7,16)(13,15)(17,21,23,18)(19,28,30,24)(20,31,33,25)(22,26,35,34)(27,39,41,36)(29,37,43,42)(32,38,45,44)(40,46,48,47);(17,22,23,35)(18,26,21,34)(19,29,30,43)(20,32,33,45)(24,37,28,42)(25,38,31,44)(27,40,41,48)(36,46,39,47)

Polycyclic presentation:

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

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, 6341926);