#148097

order	512
nilpotency class	4
isoclinism	isoclinic class of #136733
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)(5,6,10,15,11,17,35,16)(7,12,21,39,23,40,22,49)(8,24,27,46,28,61,34,60)(9,29,32,14,33,47,58,48)(13,31,42,26,44,62,43,63)(18,50,52,67,54,41,53,66)(19,45,56,38,20,57,30,37)(25,36,59,64,51,65,55,68);(1,2,3,4)(5,7)(6,12)(8,25)(9,30)(10,21)(11,23)(13,41)(14,45)(15,39)(16,49)(17,40)(18,26)(19,32)(20,58)(22,35)(24,36)(27,59)(28,51)(29,37)(31,53)(33,56)(34,55)(38,47)(42,66)(43,67)(44,50)(46,64)(48,57)(52,62)(54,63)(60,68)(61,65);(5,8,11,28)(6,13,17,44)(7,18,23,54)(9,31,33,62)(10,34,35,27)(12,36,40,65)(14,46,48,60)(15,43,16,42)(19,55,20,59)(21,53,22,52)(24,29,61,47)(25,56,51,30)(26,32,63,58)(37,66,38,67)(39,68,49,64)(41,57,50,45)

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=fh,h^a=hi,c^b=cf,e^b=eh,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, 55293);