#502380

order	512
nilpotency class	4
isoclinism	isoclinic class of #500736
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	#25321
prime radical	#33594	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=fh,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,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409576);