#148129

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2065
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

682 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)(5,6,9,18,12,19,35,15)(7,20,22,13,26,41,25,42)(8,27,29,50,33,53,32,46)(10,36,11,16,34,17,37,51)(14,43,47,65,48,31,45,30)(21,55,56,66,58,44,61,68)(23,39,59,49,60,52,24,40)(28,38,63,62,64,67,57,54);(1,2,3,4)(5,7,9,22,12,26,35,25)(6,13,15,20,19,42,18,41)(8,28,29,63,33,64,32,57)(10,24,11,23,34,59,37,60)(14,44,45,66,48,55,47,68)(16,49,36,39,51,40,17,52)(21,30,56,43,58,65,61,31)(27,62,46,38,53,54,50,67);(5,8)(6,14)(7,21)(9,29)(10,31)(11,30)(12,33)(13,38)(15,45)(16,50)(17,53)(18,47)(19,48)(20,54)(22,56)(23,28)(24,57)(25,61)(26,58)(27,36)(32,35)(34,43)(37,65)(39,55)(40,68)(41,62)(42,67)(44,52)(46,51)(49,66)(59,63)(60,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=h,g^2=i,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=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55325);