#6350356

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2645 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,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, 6257552);