#449093

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15930
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

781 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,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, 356289);