#6354643

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#121
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1491 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,d^c=dh,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, 6261839);