#6351907

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

2734 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=h,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, 6259103);