#146320

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂²	C₄²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,g^2=h,h^2=i,b^a=bd,d^a=di,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53516);