#474504

order	512
nilpotency class	3
isoclinism	isoclinic class of #454389
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2815
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16982
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

696 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)(5,26)(6,13)(7,12)(8,14)(9,18)(10,25)(11,30)(16,36)(17,33)(19,35)(20,34)(22,53)(23,54)(24,38)(27,52)(28,42)(29,41)(31,39)(32,50)(37,58)(40,55)(43,48)(44,59)(45,51)(46,62)(47,56)(49,61)(57,64)(60,63);(1,3)(2,9,14,34)(4,22,25,50)(5,17)(6,18)(7,20)(8,19)(10,36)(11,33,42,61)(12,15,13,35)(16,24,45,31)(21,51)(23,46,55,63)(26,56,41,43)(27,57,58,44)(28,47)(29,49)(30,48)(32,38)(37,62)(39,53)(40,59)(52,60)(54,64);(1,4,8,25)(2,10)(3,16,19,45)(5,27,29,58)(6,31,7,24)(9,32)(11,40)(12,39)(13,38)(14,21)(15,36)(17,46,49,63)(18,50,20,22)(23,30,55,28)(26,52)(33,60)(34,53)(35,51)(37,41)(42,54)(43,59)(44,48,57,47)(56,64)(61,62);(1,5,6,28,8,29,7,30)(2,11,12,41,14,42,13,26)(3,17,18,47,19,49,20,48)(4,23,24,27,25,55,31,58)(9,33,15,43,34,61,35,56)(10,37,38,40,21,52,39,54)(16,44,22,46,45,57,50,63)(32,59,51,60,53,64,36,62)

Polycyclic presentation:

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

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, 381700);