#6386131

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2899 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

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, 6293327);