#6352023

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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

2683 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259219);