#6351918

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	#63058	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^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, 6259114);