#148153

order	512
nilpotency class	4
isoclinism	isoclinic class of #135930
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,d^c=di,e^c=ehi,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, 55349);