#418861

order	512
nilpotency class	3
isoclinism	isoclinic class of #418860
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15266
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1878 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287080, #72287286, #72287292
maximal normal subgroup of	#72287080, #72287286, #72287292

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326057);