#6350977

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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

2588 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)(3,12)(4,9)(5,11)(6,19)(7,10)(8,14)(13,26)(15,23)(16,25)(17,24)(18,29)(20,22)(21,28)(27,31)(30,32)(33,34)(35,46)(36,51)(37,43)(38,44)(39,45)(40,49)(41,52)(42,54)(47,60)(48,58)(50,57)(53,59)(55,63)(56,62)(61,64);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,27)(17,28)(18,30)(24,31)(25,32)(26,29)(33,35,39,50)(34,40,45,58)(36,47,54,63)(37,48,38,49)(41,55,59,60)(42,56,51,64)(43,46,44,57)(52,61,53,62);(1,4,7,16)(2,9,10,25)(3,13,15,27)(5,17,6,18)(8,21,22,32)(11,24,19,29)(12,26,23,31)(14,28,20,30)(33,36,39,54)(34,41,45,59)(35,47,50,63)(37,52,38,53)(40,55,58,60)(42,44,51,43)(46,56,57,64)(48,61,49,62);(2,10)(4,16)(8,22)(11,19)(12,23)(13,27)(17,18)(28,30)(33,36)(34,42)(35,47)(37,53)(38,52)(39,54)(40,56)(41,43)(44,59)(45,51)(46,55)(48,62)(49,61)(50,63)(57,60)(58,64);(1,5,7,6)(2,11,10,19)(3,14,15,20)(4,17,16,18)(8,23,22,12)(9,24,25,29)(13,28,27,30)(21,31,32,26)(33,37,39,38)(34,43,45,44)(35,48,50,49)(36,52,54,53)(40,46,58,57)(41,42,59,51)(47,61,63,62)(55,56,60,64)

Polycyclic presentation:

2^9|b^2=i,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258173);