#6356507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356405
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	#60992	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1458 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)(3,11,13,25)(4,8,15,22)(5,17,18,9)(7,19,21,14)(12,24,26,32)(16,29,28,23)(20,30,31,27)(33,34)(35,44)(36,42)(37,43)(38,40)(39,51)(41,46)(45,59)(47,56)(48,57)(49,54)(50,58)(52,53)(55,62)(60,63)(61,64);(1,3,5,14,6,13,18,19)(2,7,9,11,10,21,17,25)(4,12,16,27,15,26,28,30)(8,20,23,24,22,31,29,32)(34,40)(41,53)(42,54)(43,51)(44,56)(55,63)(57,58)(59,64);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,24)(13,26)(14,27)(17,29)(18,28)(19,30)(21,31)(25,32)(33,35)(34,41)(36,45)(37,46)(38,47)(39,52)(40,53)(42,55)(43,56)(44,51)(48,60)(49,61)(50,62)(54,63)(57,59)(58,64);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(24,32)(27,30)(33,36,38,49)(34,42,40,54)(35,45,47,61)(37,50,39,48)(41,55,53,63)(43,58,51,57)(44,59,56,64)(46,62,52,60);(3,13)(7,21)(11,25)(12,26)(14,19)(20,31)(24,32)(27,30)(33,37,38,39)(34,43,40,51)(35,46,47,52)(36,48,49,50)(41,56,53,44)(42,57,54,58)(45,60,61,62)(55,59,63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263703);