#458563

order	512
nilpotency class	3
isoclinism	isoclinic class of #453994
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 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)(5,25,28,41)(6,12)(7,13)(8,14)(9,18,33,20)(11,29,42,27)(16,35,46,43)(17,47)(21,31,52,60)(22,40,54,62)(23,37)(24,39)(26,57,55,36)(30,38)(32,51)(44,49)(45,64)(48,63)(50,56)(53,61)(58,59);(1,3,6,20,8,19,7,18)(2,9,12,34,14,33,13,15)(4,21,23,16,24,52,30,46)(5,17,29,49,28,51,27,50)(10,35,37,60,39,43,38,31)(11,32,25,56,42,47,41,44)(22,53,55,45,54,48,26,58)(36,61,62,64,57,63,40,59);(1,4)(2,10)(3,16,19,46)(5,26,28,55)(6,23)(7,30)(8,24)(9,31,33,60)(11,40,42,62)(12,37)(13,38)(14,39)(15,43,34,35)(17,48)(18,21,20,52)(22,27,54,29)(25,36,41,57)(32,61)(44,59)(45,50)(47,63)(49,58)(51,53)(56,64);(1,5)(2,11)(3,17)(4,22)(6,29)(7,27)(8,28)(9,32)(10,36)(12,25)(13,41)(14,42)(15,44)(16,45)(18,50)(19,51)(20,49)(21,53)(23,55)(24,54)(26,30)(31,59)(33,47)(34,56)(35,61)(37,62)(38,40)(39,57)(43,63)(46,58)(48,52)(60,64)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,e^2=i,g^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,g^b=gi,h^b=hi,d^c=de,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, 365759);