#6529590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
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

2650 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,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, 6436786);