#499045

order	512
nilpotency class	4
isoclinism	isoclinic class of #498802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2034
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1045 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)(3,13)(4,18)(5,22)(6,28)(7,11)(8,15)(9,20)(10,26)(12,25)(14,41)(16,46)(17,32)(19,52)(21,55)(23,57)(24,39)(27,56)(29,37)(30,54)(31,59)(33,43)(34,53)(35,60)(36,49)(38,47)(40,45)(42,51)(44,62)(48,61)(50,64)(58,63);(1,3,6,17,7,15,25,45)(2,8,12,32,11,13,28,40)(4,14,21,44,19,42,29,50)(5,23,26,59,27,16,47,49)(9,33,35,63,34,30,39,61)(10,36,22,46,38,31,56,57)(18,51,37,62,52,41,55,64)(20,54,24,58,53,43,60,48);(1,4)(2,9)(3,14,15,42)(5,24)(6,29)(7,19)(8,30,13,33)(10,37)(11,34)(12,39)(16,48,23,58)(17,50,45,44)(18,22)(20,26)(21,25)(27,60)(28,35)(31,62,36,64)(32,63,40,61)(38,55)(41,57,51,46)(43,59,54,49)(47,53)(52,56);(4,19)(9,34)(14,42)(18,52)(20,53)(21,29)(24,60)(30,33)(35,39)(37,55)(41,51)(43,54)(44,50)(48,58)(61,63)(62,64)

Polycyclic presentation:

2^9|b^2=h,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di,f^c=fh,g^c=gi,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, 406241);