#511769

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
prime radical	#63091	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3100 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 418965);