#511860

order	512
nilpotency class	4
isoclinism	isoclinic class of #511859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32899
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2020 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)(3,14)(4,19)(5,23)(6,9)(7,13)(8,16)(10,22)(11,26)(12,28)(15,43)(17,40)(18,38)(20,36)(21,37)(24,33)(25,35)(27,42)(29,39)(30,47)(31,51)(32,48)(34,55)(41,46)(44,49)(45,62)(50,53)(52,60)(54,61)(56,59)(57,64)(58,63);(1,3,7,18)(2,8,13,33)(4,15,22,46)(5,24,26,16)(6,27,28,17)(9,30,12,32)(10,31,19,49)(11,38,23,14)(20,51,53,44)(21,54,55,45)(25,47,29,48)(34,57,37,58)(35,40,39,42)(36,43,50,41)(52,63,56,64)(59,61,60,62);(2,9)(6,28)(8,30)(10,34)(11,39)(12,13)(14,40)(17,27)(19,37)(21,55)(23,35)(25,29)(31,57)(32,33)(36,60)(38,42)(41,61)(43,62)(45,54)(47,48)(49,58)(50,59)(52,56)(63,64);(1,4)(2,10,13,19)(3,15)(5,20)(6,21)(7,22)(8,31,33,49)(9,34,12,37)(11,36,23,50)(14,41,38,43)(16,44)(17,45)(18,46)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57,32,58)(35,59,39,60)(40,61,42,62)(47,63)(48,64)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,g^d=gi,f^e=fi

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, 419056);