#6351920

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	#63067	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,8,13)(3,14,17,19)(4,16,9,23)(5,10,20,25)(6,12,7,11)(15,28,29,31)(18,30,24,32)(21,27,22,26)(33,34,39,36)(35,43,47,56)(37,48,50,41)(38,49,53,42)(40,52,44,46)(45,59,60,57)(51,55,63,62)(54,61,58,64);(1,3)(2,9)(4,13)(5,15)(6,23)(7,16)(8,17)(10,24)(11,19)(12,14)(18,25)(20,29)(21,32)(22,30)(26,31)(27,28)(33,35,39,47)(34,40,36,44)(37,45,50,60)(38,52,53,46)(41,54,48,58)(42,56,49,43)(51,64,63,61)(55,59,62,57);(1,4,8,9)(2,3,13,17)(5,18,20,24)(6,19,7,14)(10,15,25,29)(11,16,12,23)(21,31,22,28)(26,30,27,32)(33,36,39,34)(35,44,47,40)(37,48,50,41)(38,42,53,49)(43,52,56,46)(45,58,60,54)(51,55,63,62)(57,64,59,61);(1,5)(2,10)(3,15)(4,18)(6,21)(7,22)(8,20)(9,24)(11,26)(12,27)(13,25)(14,28)(16,30)(17,29)(19,31)(23,32)(33,37)(34,41)(35,45)(36,48)(38,51)(39,50)(40,54)(42,55)(43,57)(44,58)(46,61)(47,60)(49,62)(52,64)(53,63)(56,59);(5,20)(10,25)(15,29)(18,24)(21,22)(26,27)(28,31)(30,32)(37,50)(41,48)(45,60)(51,63)(54,58)(55,62)(57,59)(61,64)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^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, 6259116);