#6351942

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

2609 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^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, 6259138);