#458561

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16320
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

961 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)(3,15,19,34)(4,10)(5,25,28,39)(6,12)(7,14)(8,13)(9,18,33,20)(11,29,40,27)(16,44,21,42)(17,47)(22,38,54,61)(23,36)(24,37)(26,57,55,35)(30,41)(31,52,59,46)(32,50)(43,49)(45,63)(48,62)(51,56)(53,58)(60,64);(1,3,6,20,8,19,7,18)(2,9,12,34,13,33,14,15)(4,21,23,46,24,16,30,52)(5,17,29,49,28,50,27,51)(10,31,36,44,37,59,41,42)(11,32,25,56,40,47,39,43)(22,53,55,64,54,45,26,48)(35,58,61,60,57,63,38,62);(1,4)(2,10)(3,16,19,21)(5,26,28,55)(6,23)(7,30)(8,24)(9,31,33,59)(11,38,40,61)(12,36)(13,37)(14,41)(15,42,34,44)(17,48)(18,46,20,52)(22,27,54,29)(25,35,39,57)(32,60)(43,58)(45,51)(47,62)(49,53)(50,64)(56,63);(1,5)(2,11)(3,17)(4,22)(6,29)(7,27)(8,28)(9,32)(10,35)(12,25)(13,40)(14,39)(15,43)(16,45)(18,51)(19,50)(20,49)(21,53)(23,55)(24,54)(26,30)(31,58)(33,47)(34,56)(36,61)(37,57)(38,41)(42,62)(44,60)(46,64)(48,52)(59,63)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 365757);