#6532519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

2850 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,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, 6439715);