#6394059

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	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2730 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,26,21,25,34,39)(19,27,30,43,29,23,36,38)(20,31,33,41,32,24,35,40)(28,37,45,47,44,42,46,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,21,29)(18,23,25,27)(20,28,32,44)(22,30,34,36)(24,37,31,42)(26,38,39,43)(33,45,35,46)(40,47,41,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,22,35,21,32,34,33)(18,24,39,40,25,31,26,41)(19,28,30,46,29,44,36,45)(23,37,43,47,27,42,38,48);(3,12)(7,10)(11,13)(15,16)(19,29)(23,27)(28,44)(30,36)(37,42)(38,43)(45,46)(47,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,25)(19,29)(24,31)(26,39)(28,44)(30,36)(40,41)(45,46)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,e^2=fi,h^2=i,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, 6301255);