#6355290

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355203
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62102	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262486);