#6350372

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

2647 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,e^2=i,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, 6257568);