#6343209

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62762	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2686 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)(3,12)(4,9)(5,10)(6,19)(7,11)(8,14)(13,27)(15,22)(16,24)(17,25)(18,26)(20,23)(21,28)(29,32)(30,31)(33,34,37,41,38,42,48,54)(35,39,45,51,44,50,57,61)(36,40,47,53,46,52,58,62)(43,49,56,60,55,59,63,64);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,28)(17,29)(18,30)(24,31)(25,27)(26,32)(33,35)(34,39)(36,43)(37,45)(38,44)(40,49)(41,51)(42,50)(46,55)(47,56)(48,57)(52,59)(53,60)(54,61)(58,63)(62,64);(1,4)(2,9)(3,13)(5,16)(6,17)(7,18)(8,21)(10,24)(11,26)(12,27)(14,28)(15,30)(19,25)(20,29)(22,31)(23,32)(33,36)(34,40)(35,43)(37,47)(38,46)(39,49)(41,53)(42,52)(44,55)(45,56)(48,58)(50,59)(51,60)(54,62)(57,63)(61,64);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,18,17)(8,22,23,12)(9,24,26,25)(13,28,30,29)(21,31,32,27)(36,46)(40,52)(43,55)(47,58)(49,59)(53,62)(56,63)(60,64);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,17,18,16)(8,22,23,12)(9,25,26,24)(13,29,30,28)(21,27,32,31)(35,44)(39,50)(43,55)(45,57)(49,59)(51,61)(56,63)(60,64)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,e^b=ei,f^b=fh,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, 6250405);