#512675

order	512
nilpotency class	5
isoclinism	isoclinic class of #512663
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	#60807	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

794 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)(4,9,21,34,20,35,49,62)(6,12,27,39,28,10,24,37)(8,23,31,26,30,45,57,18)(15,41,42,60,19,46,47,63)(22,33,53,59,50,36,52,61)(29,54,56,48,55,51,64,43);(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)(33,41)(34,60)(35,55)(36,46)(39,57)(40,58)(42,52)(47,53)(48,50)(49,54)(56,61)(59,64)(62,63);(1,4)(2,9)(3,15)(5,21)(6,22)(7,20)(8,29)(10,36)(11,34)(12,33)(13,35)(14,41)(16,19)(17,42)(18,43)(23,54)(24,52)(25,49)(26,48)(27,53)(28,50)(30,55)(31,56)(32,46)(37,61)(38,60)(39,59)(40,62)(44,47)(45,51)(57,64)(58,63);(3,16)(8,30)(14,32)(15,19)(17,44)(18,26)(23,45)(29,55)(31,57)(38,58)(41,46)(42,47)(43,48)(51,54)(56,64)(60,63)

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,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, 419871);