#6352021

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

2623 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)(19,29)(20,32)(21,25)(22,26)(23,27)(24,28)(30,43)(31,35)(33,39)(34,44)(36,40)(37,41)(38,42)(45,47)(46,48);(1,2)(3,7)(4,12)(5,8)(6,11)(9,15)(10,14)(13,16)(17,19)(18,20)(21,30)(22,31)(23,32)(24,33)(25,34)(26,29)(27,39)(28,35)(36,44)(37,46)(38,45)(40,47)(41,43)(42,48);(4,11)(6,12)(10,16)(13,14)(17,20)(18,19)(21,34)(22,29)(23,39)(24,35)(25,30)(26,31)(27,32)(28,33)(36,47)(37,43)(38,48)(40,44)(41,46)(42,45);(1,3,5,9)(2,7,8,15)(4,10,11,16)(6,13,12,14)(17,21,24,38)(18,25,28,42)(19,30,33,45)(20,34,35,48)(22,37,23,36)(26,41,27,40)(29,43,39,47)(31,46,32,44);(3,9)(7,15)(10,16)(13,14)(17,22,24,23)(18,26,28,27)(19,31,33,32)(20,29,35,39)(21,36,38,37)(25,40,42,41)(30,44,45,46)(34,47,48,43)

Polycyclic presentation:

2^9|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, 6259217);