#6394048

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342866
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	C₂⁴	C₂⁵
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2876 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,27)(20,24,32,38)(21,25)(22,26)(23,30)(28,41,43,35)(29,39)(31,37,45,48)(33,36)(34,40)(42,47,44,46);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,23)(20,28)(21,33)(22,30)(24,35)(25,39)(26,27)(29,34)(31,44)(32,43)(36,40)(37,47)(38,41)(42,45)(46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,22,32)(18,24)(19,28,30,43)(21,31,34,45)(23,35)(25,37)(26,38)(27,41)(29,42,33,44)(36,46)(39,47)(40,48);(3,12)(7,10)(11,13)(15,16)(17,21,22,34)(18,25,26,40)(19,29,30,33)(20,31,32,45)(23,36,27,39)(24,37,38,48)(28,42,43,44)(35,46,41,47);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,26)(19,30)(24,38)(25,40)(28,43)(29,33)(37,48)(42,44)

Polycyclic presentation:

2^9|a^2=h,c^2=fgh,d^2=i,e^2=fi,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,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, 6301244);