#499313

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#466
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,9,16,13,20,12,36)(7,21,23,42,27,41,26,58)(8,28,37,51,33,46,30,47)(10,19,34,18,35,50,11,17)(14,38,40,57,43,25,52,24)(15,32,48,31,49,64,45,63)(22,53,59,65,56,39,54,67)(29,60,55,68,62,66,61,44);(1,2)(3,4)(5,7,11,25,12,26,35,57,13,27,34,38,9,23,10,24)(6,14,19,42,16,40,18,41,20,43,50,58,36,52,17,21)(8,29,32,54,30,61,63,56,33,62,64,59,37,55,31,22)(15,44,47,67,48,60,28,53,49,68,51,65,45,66,46,39);(2,4)(5,8)(6,15)(7,22)(9,30)(10,32)(11,31)(12,37)(13,33)(14,39)(16,45)(17,47)(18,51)(19,46)(20,49)(21,44)(23,54)(24,29)(25,55)(26,59)(27,56)(28,50)(34,63)(35,64)(36,48)(38,61)(40,65)(41,68)(42,66)(43,53)(52,67)(57,62)(58,60);(5,9,13,12)(6,16,20,36)(7,23,27,26)(8,30,33,37)(10,34,35,11)(14,40,43,52)(15,45,49,48)(17,19,18,50)(21,42,41,58)(22,54,56,59)(24,38,57,25)(28,47,46,51)(29,61,62,55)(31,32,63,64)(39,65,53,67)(44,66,68,60)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=i,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^c=eh,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406509);