#512680

order	512
nilpotency class	5
isoclinism	isoclinic class of #512666
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#60823	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ehi,f^b=fgh,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419876);