#511767

order	512
nilpotency class	4
isoclinism	isoclinic class of #511767
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32878
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3108 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)(3,11)(4,20)(5,24)(6,14)(7,13)(8,15)(9,17)(10,22)(12,27)(16,44)(18,43)(19,39)(21,53)(23,42)(25,37)(26,41)(28,40)(29,36)(30,38)(31,59)(32,57)(33,49)(34,46)(35,56)(45,50)(47,51)(48,62)(52,55)(54,60)(58,63)(61,64);(1,3)(2,9)(4,16)(5,25)(6,18)(7,30)(8,19)(10,31)(11,24)(12,38)(13,32)(14,37)(15,33)(17,27)(20,50)(21,54)(22,46)(23,47)(26,49)(28,57)(29,48)(34,52)(35,51)(36,60)(39,40)(41,43)(42,61)(44,53)(45,55)(56,59)(58,64)(62,63);(1,4)(2,10)(3,16)(5,21)(6,29)(7,22)(8,23)(9,31)(11,34)(12,35)(13,42)(14,20)(15,36)(17,45)(18,48)(19,47)(24,52)(25,54)(26,58)(27,55)(28,56)(30,46)(32,61)(33,60)(37,50)(38,51)(39,62)(40,63)(41,53)(43,44)(49,64)(57,59);(1,3)(2,11)(4,16)(5,17)(6,30)(7,18)(8,19)(9,24)(10,34)(12,37)(13,43)(14,38)(15,39)(20,51)(21,45)(22,48)(23,47)(25,27)(26,57)(28,49)(29,46)(31,52)(32,41)(33,40)(35,50)(36,62)(42,44)(53,61)(54,55)(56,64)(58,59)(60,63)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280969
maximal normal subgroup of	#72280969

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418963);