#6571609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2837 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:

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

Polycyclic presentation:

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

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, 6478805);