#6489617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2680 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,24,20)(19,28,32,40)(21,26,36,33)(22,27,23,34)(25,38,29,31)(30,43,47,48)(35,41,37,42)(39,45,44,46);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,19)(18,25)(20,29)(21,30)(22,31)(23,38)(24,32)(26,39)(27,40)(28,34)(33,44)(35,45)(36,47)(37,46)(41,43)(42,48);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(17,20,24,18)(19,29,32,25)(21,33,36,26)(22,34,23,27)(28,38,40,31)(30,44,47,39)(35,42,37,41)(43,45,48,46);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14)(17,21)(18,26)(19,30)(20,33)(22,37)(23,35)(24,36)(25,39)(27,42)(28,43)(29,44)(31,46)(32,47)(34,41)(38,45)(40,48);(2,10)(7,16)(8,9)(14,15)(17,22,24,23)(18,27,20,34)(19,31,32,38)(21,35,36,37)(25,40,29,28)(26,41,33,42)(30,45,47,46)(39,43,44,48)

Polycyclic presentation:

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

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, 6396813);