#6434723

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3002 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 3002 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) = 64

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,b^a=bf,c^a=cg,d^a=dh,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, 6341919);