#6350347

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

2643 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,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, 6257543);