#418862

order	512
nilpotency class	3
isoclinism	isoclinic class of #418851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2679
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15266
prime radical	#60366	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1887 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)(3,15)(4,21)(5,27,31,41)(6,13)(7,12)(8,14)(9,18)(10,25)(11,30,42,29)(16,44)(17,48,49,33)(19,35)(20,34)(22,52)(23,37,55,61)(24,39)(26,38)(28,40,50,43)(32,47)(36,46)(45,62,64,59)(51,54,57,56)(53,58,63,60);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,36)(11,40)(12,34)(13,15)(14,35)(16,24)(17,29)(21,32)(23,45)(25,47)(26,46)(27,33)(30,49)(31,50)(37,60)(38,44)(39,52)(41,48)(42,43)(51,59)(53,54)(55,64)(56,63)(57,62)(58,61);(1,4)(2,10)(3,16)(5,23)(6,25)(7,24)(8,26)(9,32)(11,37)(12,21)(13,38)(14,39)(15,36)(17,45)(18,46)(19,22)(20,47)(27,57)(28,53)(29,56)(30,54)(31,55)(33,58)(34,52)(35,44)(40,62)(41,51)(42,61)(43,59)(48,60)(49,64)(50,63);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,37)(12,41)(13,27)(14,42)(15,43)(16,45)(18,50)(19,28)(20,49)(21,51)(22,53)(24,56)(25,54)(26,55)(32,58)(34,40)(35,48)(36,59)(38,57)(39,61)(44,60)(46,63)(47,64)(52,62)

Polycyclic presentation:

2^9|a^2=ef,e^2=i,f^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,f^c=fi,g^c=gi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326058);