#506392

order	512
nilpotency class	4
isoclinism	isoclinic class of #501994
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28227
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

593 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)(3,12)(4,21)(5,28)(6,32)(7,35)(8,14)(9,15)(10,17)(11,25)(13,29)(16,47)(18,52)(19,45)(20,46)(22,55)(23,53)(24,42)(26,44)(27,43)(30,39)(31,38)(33,36)(34,61)(37,58)(40,49)(41,50)(48,54)(51,64)(56,62)(57,63)(59,60);(1,3,6,17,8,19,33,31,9,20,34,30,7,18,29,5)(2,10,36,46,35,28,32,45,15,39,13,12,14,38,61,52)(4,22,56,58,25,16,48,57,27,50,24,23,26,51,59,49)(11,40,42,47,21,53,54,55,44,63,62,64,43,37,60,41);(1,4)(2,11)(3,16)(5,23)(6,24)(7,26)(8,25)(9,27)(10,37)(12,41)(13,42)(14,43)(15,44)(17,49)(18,22)(19,50)(20,51)(21,35)(28,40)(29,48)(30,57)(31,58)(32,60)(33,59)(34,56)(36,62)(38,63)(39,53)(45,64)(46,55)(47,52)(54,61);(1,5,29,18,7,30,34,20,9,31,33,19,8,17,6,3)(2,12,32,10,14,45,36,38,15,46,61,39,35,52,13,28)(4,23,48,22,26,57,56,51,27,58,59,50,25,49,24,16)(11,41,60,37,43,64,62,63,44,55,54,53,21,47,42,40)

Polycyclic presentation:

2^9|b^2=egh,d^2=eh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413588);