#6529469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
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	#3455	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2648 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi,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, 6436665);