#148146

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
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	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 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)(7,20)(8,26)(9,18)(10,33)(11,15)(12,19)(13,22)(14,29)(16,36)(17,32)(21,54)(23,60)(24,40)(25,52)(27,63)(28,49)(30,51)(31,44)(34,48)(35,43)(37,47)(38,41)(39,62)(42,58)(45,64)(46,65)(50,59)(53,56)(55,68)(57,67)(61,66);(5,7,9,22,12,25,11,38)(6,13,15,20,19,41,18,52)(8,27,28,62,31,56,30,61)(10,34,32,24,37,59,36,23)(14,42,43,67,46,68,45,54)(16,48,47,60,17,50,33,40)(21,35,55,29,57,64,58,65)(26,39,51,63,44,66,49,53);(5,8,12,31)(6,14,19,46)(7,21,25,57)(9,30,11,28)(10,35,37,64)(13,39,41,66)(15,45,18,43)(16,49,17,51)(20,53,52,63)(22,58,38,55)(23,61,24,62)(26,47,44,33)(27,34,56,59)(29,32,65,36)(40,67,60,54)(42,50,68,48)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=hi,f^2=hi,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=eh,f^c=fh,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, 55342);