#6348954

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62762	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2625 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)(5,10)(6,19)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,29)(20,23)(21,28)(27,32)(30,31)(33,34,39,44)(35,40,47,55)(36,48,50,58)(37,43,38,42)(41,52,56,49)(45,59,61,63)(46,57,51,54)(53,60,64,62);(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)(34,40)(36,45)(37,46)(38,51)(39,47)(41,53)(42,54)(43,57)(44,55)(48,63)(49,60)(50,61)(52,62)(56,64)(58,59);(1,4)(2,9)(3,13)(5,17)(6,18)(7,16)(8,21)(10,25)(11,24)(12,26)(14,28)(15,27)(19,29)(20,30)(22,32)(23,31)(33,36)(34,41)(35,45)(37,49)(38,52)(39,50)(40,53)(42,48)(43,58)(44,56)(46,60)(47,61)(51,62)(54,63)(55,64)(57,59);(4,16)(9,24)(13,27)(17,18)(21,31)(25,29)(26,32)(28,30)(33,37,39,38)(34,42,44,43)(35,46,47,51)(36,49,50,52)(40,54,55,57)(41,48,56,58)(45,60,61,62)(53,63,64,59);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,17,16,18)(8,22,23,12)(9,25,24,29)(13,28,27,30)(21,32,31,26)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256150);