#148155

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

670 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)(7,19)(8,26)(9,32)(10,36)(11,16)(12,18)(13,21)(14,29)(15,35)(17,34)(20,55)(22,41)(23,42)(24,44)(25,43)(27,40)(28,47)(30,49)(31,52)(33,50)(37,64)(38,54)(39,51)(45,59)(46,63)(48,62)(53,57)(56,68)(58,66)(60,65)(61,67);(5,7,9,21,12,25,35,24)(6,13,15,19,18,44,32,43)(8,27,28,57,31,61,33,60)(10,37,11,38,34,23,39,22)(14,45,46,68,48,55,49,66)(16,42,36,54,51,64,17,41)(20,29,56,30,59,62,58,63)(26,53,50,40,52,65,47,67);(5,8,12,31)(6,14,18,48)(7,20,25,59)(9,33,35,28)(10,30,34,63)(11,29,39,62)(13,40,44,67)(15,49,32,46)(16,50,51,47)(17,52,36,26)(19,53,43,65)(21,58,24,56)(22,27,38,61)(23,57,37,60)(41,55,54,45)(42,68,64,66)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fhi,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, 55351);