#468073

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16600
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1067 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)(4,20)(5,24)(6,14)(7,12)(8,13)(9,18)(10,23)(11,28)(16,44)(17,32)(19,41)(21,54)(22,36)(25,53)(26,39)(27,38)(29,40)(30,33)(31,46)(34,48)(35,56)(37,55)(42,47)(43,50)(45,61)(49,58)(51,60)(52,59)(57,64)(62,63);(1,3)(2,9,13,33)(4,16)(5,17)(6,18)(7,30)(8,19)(10,34,20,31)(11,32,39,60)(12,15,14,41)(21,45)(22,46)(23,47)(24,43,38,59)(25,49)(26,50)(27,51)(28,52)(29,48)(35,61,53,57)(36,42,40,44)(37,62,54,58)(55,64)(56,63);(1,4)(2,10,13,20)(3,16)(5,25)(6,29)(7,22)(8,23)(9,31,33,34)(11,37,39,54)(12,40,14,36)(15,42,41,44)(17,49)(18,48)(19,47)(21,26)(24,53,38,35)(27,56)(28,55)(30,46)(32,58,60,62)(43,61,59,57)(45,50)(51,63)(52,64);(1,5)(2,11)(3,17)(4,21)(6,28)(7,26)(8,27)(9,32)(10,35)(12,24)(13,39)(14,38)(15,43)(16,45)(18,52)(19,51)(20,53)(22,56)(23,55)(25,29)(30,50)(31,57)(33,60)(34,61)(36,54)(37,40)(41,59)(42,58)(44,62)(46,63)(47,64)(48,49)

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,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, 375269);