#6434729

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×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2888 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 2888 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)(21,32)(23,34)(25,39)(28,43)(30,33)(35,45)(37,40)(41,44);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,35)(28,36)(29,34)(30,37)(31,38)(32,39)(33,40)(41,47)(42,45)(43,46)(44,48);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,35)(25,36)(26,34)(30,41)(31,42)(32,43)(33,44)(37,47)(38,45)(39,46)(40,48);(2,8)(6,14)(7,16)(13,15)(17,20,22,31)(18,24,26,38)(19,27,29,42)(21,33,32,30)(23,35,34,45)(25,40,39,37)(28,44,43,41)(36,48,46,47);(17,21,22,32)(18,25,26,39)(19,28,29,43)(20,30,31,33)(23,36,34,46)(24,37,38,40)(27,41,42,44)(35,47,45,48)

Polycyclic presentation:

2^9|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, 6341925);