#148109

order	512
nilpotency class	4
isoclinism	isoclinic class of #136735
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
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

672 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,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, 55305);