#6386306

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342800
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2882 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293502);