#501737

order	512
nilpotency class	4
isoclinism	isoclinic class of #500737
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24812
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=e,d^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=di,f^c=fi,g^c=gh,h^c=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, 408933);