#6344205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343524
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	#62839	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2664 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bf,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251401);