#6436616

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432876
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2881 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)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,18,22,25)(19,27,30,43)(20,24,33,41)(21,34,37,26)(23,35,39,29)(28,46,31,45)(32,44,36,42)(38,48,40,47);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,23)(20,31)(21,35)(22,30)(24,40)(25,39)(26,43)(27,34)(28,33)(29,37)(32,47)(36,48)(38,41)(42,46)(44,45);(3,9)(6,12)(10,15)(13,16)(17,20,22,33)(18,24,25,41)(19,28,30,31)(21,36,37,32)(23,38,39,40)(26,44,34,42)(27,45,43,46)(29,48,35,47);(2,7)(6,12)(8,14)(13,16)(18,25)(23,39)(24,41)(26,34)(27,43)(38,40)(42,44)(45,46);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)(18,25)(23,39)(24,41)(26,34)(27,43)(38,40)(42,44)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343812);