#6351906

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
prime radical	#63049	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2758 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)(17,18,24,29)(19,30,25,39)(20,32,35,33)(21,26,38,43)(22,27,23,28)(31,44,40,47)(34,46,48,45)(36,41,37,42);(4,13)(8,12)(9,16)(11,14)(17,19,24,25)(18,20,29,35)(21,31,38,40)(22,32,23,33)(26,34,43,48)(27,30,28,39)(36,45,37,46)(41,47,42,44);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20)(18,25)(19,29)(21,34)(22,30)(23,39)(24,35)(26,40)(27,33)(28,32)(31,43)(36,47)(37,44)(38,48)(41,46)(42,45);(2,7)(6,15)(8,12)(9,16)(17,21)(18,26)(19,31)(20,34)(22,37)(23,36)(24,38)(25,40)(27,42)(28,41)(29,43)(30,44)(32,46)(33,45)(35,48)(39,47);(17,22,24,23)(18,27,29,28)(19,32,25,33)(20,30,35,39)(21,36,38,37)(26,41,43,42)(31,45,40,46)(34,47,48,44)

Polycyclic presentation:

2^9|a^2=hi,b^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, 6259102);