#6434719

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2981 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 2981 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)(19,23)(20,24)(21,25)(22,26)(27,39)(28,34)(29,35)(30,36)(31,37)(32,38)(33,41)(40,47)(42,45)(43,46)(44,48);(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)(17,21,22,32)(18,25,26,38)(19,28,29,42)(20,30,31,43)(23,34,35,45)(24,36,37,46)(27,40,41,48)(33,44,39,47);(18,26)(23,35)(24,37)(25,38)(33,39)(34,45)(36,46)(44,47)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,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, 6341915);