#460458

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16364
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

907 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367654);