#6351916

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	#63054	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2600 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:

(1,2,7,12)(3,13,16,26)(4,9,17,23)(5,11,6,10)(8,20,22,15)(14,27,28,32)(18,25,19,24)(21,30,31,29)(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)(2,8)(4,14)(5,20)(6,15)(7,16)(9,21)(10,26)(11,13)(12,22)(17,28)(18,30)(19,29)(23,31)(24,32)(25,27)(33,35,38,46)(34,39,42,54)(36,44,48,59)(37,51,52,45)(40,53,47,61)(41,56,50,43)(49,64,63,60)(55,58,62,57);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,26)(12,22)(17,28)(18,29)(19,30)(23,31)(24,27)(25,32)(37,52)(41,50)(43,56)(45,51)(49,63)(55,62)(57,58)(60,64);(1,4)(2,9)(3,14)(5,18)(6,19)(7,17)(8,21)(10,24)(11,25)(12,23)(13,27)(15,29)(16,28)(20,30)(22,31)(26,32)(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);(4,17)(9,23)(14,28)(18,19)(21,31)(24,25)(27,32)(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=i,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, 6259112);