#512681

order	512
nilpotency class	5
isoclinism	isoclinic class of #512669
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	#60831	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

788 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:

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

Polycyclic presentation:

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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419877);