#451787

order	512
nilpotency class	3
isoclinism	isoclinic class of #427841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2778
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16014
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

933 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=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, 358983);