#6350589

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
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

2754 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)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18)(19,30)(20,35)(21,27)(22,28)(23,29)(24,33)(25,36)(26,38)(31,44)(32,42)(34,41)(37,43)(39,47)(40,46)(45,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,34)(18,24,29,42)(20,31,38,47)(21,32,22,33)(25,39,43,44)(26,40,35,48)(27,30,28,41)(36,45,37,46);(17,20)(18,25)(19,31)(21,36)(22,37)(23,38)(24,39)(26,27)(28,35)(29,43)(30,40)(32,45)(33,46)(34,47)(41,48)(42,44);(2,8)(7,15)(9,13)(10,16)(17,20)(18,26)(19,31)(21,37)(22,36)(23,38)(24,40)(25,27)(28,43)(29,35)(30,39)(32,46)(33,45)(34,47)(41,44)(42,48);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,21,23,22)(18,27,29,28)(19,32,34,33)(20,36,38,37)(24,30,42,41)(25,26,43,35)(31,45,47,46)(39,40,44,48)

Polycyclic presentation:

2^9|b^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257785);