#147971

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55167);