#6393696

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	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2697 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)(17,18,22,28,23,27,21,26)(19,24,30,37,31,38,34,41)(20,25,32,39,33,40,35,42)(29,36,45,47,44,48,43,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,31)(18,24,27,38)(20,29,33,44)(21,34,22,30)(25,36,40,48)(26,41,28,37)(32,45,35,43)(39,47,42,46);(17,20,22,35,23,33,21,32)(18,25,26,39,27,40,28,42)(19,29,30,43,31,44,34,45)(24,36,41,47,38,48,37,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,26,27,28)(19,30,31,34)(20,32,33,35)(24,37,38,41)(25,39,40,42)(29,43,44,45)(36,46,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=gh,d^2=f,e^2=i,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, 6300892);