#6344286

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343678
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	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2594 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)(35,45)(36,41)(37,42)(38,43)(39,44)(40,47)(46,59)(48,54)(49,56)(50,58)(51,57)(52,55)(53,61)(60,64)(62,63);(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)(33,35,39,48)(34,40,44,55)(36,46,50,62)(37,47,38,52)(41,53,58,64)(42,54,43,45)(49,60,51,61)(56,59,57,63);(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)(35,46)(37,51)(38,49)(39,50)(40,53)(42,57)(43,56)(44,58)(45,59)(47,61)(48,62)(52,60)(54,63)(55,64);(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)(34,42,44,43)(35,47,48,52)(36,49,50,51)(40,54,55,45)(41,56,58,57)(46,60,62,61)(53,59,64,63);(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)(33,37,39,38)(34,43,44,42)(35,47,48,52)(36,49,50,51)(40,45,55,54)(41,57,58,56)(46,60,62,61)(53,63,64,59)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251482);