#6544203

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
prime radical	#62956	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2699 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,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, 6451399);