#6343958

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	#62825	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2619 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)(4,9,17,23,18,24,19,25)(14,27,29,31,30,21,28,32)(33,34)(35,45,48,54)(36,41)(37,42)(38,43)(39,44)(40,47,55,52)(46,59,62,63)(49,56)(50,58)(51,57)(53,61,64,60);(1,3,7,16)(2,8,11,13)(4,14,18,30)(5,15,6,20)(9,21,24,27)(10,22,12,26)(17,28,19,29)(23,31,25,32)(33,35)(34,40)(36,46)(37,47)(38,52)(39,48)(41,53)(42,45)(43,54)(44,55)(49,60)(50,61)(51,62)(56,63)(57,64)(58,59);(1,4)(2,9)(3,14)(5,19)(6,17)(7,18)(8,21)(10,25)(11,24)(12,23)(13,27)(15,29)(16,30)(20,28)(22,32)(26,31)(33,36)(34,41)(35,46)(37,50)(38,49)(39,51)(40,53)(42,58)(43,56)(44,57)(45,59)(47,61)(48,62)(52,60)(54,63)(55,64);(33,37,39,38)(34,42,44,43)(35,47,48,52)(36,49,51,50)(40,45,55,54)(41,56,57,58)(46,60,62,61)(53,63,64,59);(1,5,7,6)(2,10,11,12)(3,15,16,20)(4,17,18,19)(8,22,13,26)(9,23,24,25)(14,28,30,29)(21,31,27,32)(33,37,39,38)(34,43,44,42)(35,47,48,52)(36,50,51,49)(40,54,55,45)(41,56,57,58)(46,61,62,60)(53,63,64,59)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=hi,f^2=i,g^2=h,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, 6251154);