Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6)(7,19)(8,25,30,42)(9,31)(10,17)(11,16)(12,18)(13,22)(14,29,43,28)(15,34)(20,47,50,36)(21,37)(23,39)(24,38)(26,40,49,63)(27,41,58,66)(32,61)(33,46)(35,45)(44,51)(48,64,55,68)(52,59)(53,56,60,57)(54,62,67,65);(5,7,12,24)(6,13,18,39)(8,26,30,49)(9,32,35,51)(10,23,11,22)(14,40,43,63)(15,44,46,61)(16,38,17,19)(20,29,50,28)(21,34,52,33)(25,36,42,47)(27,55,58,48)(31,59,45,37)(41,65,66,62)(53,64,60,68)(54,57,67,56);(5,8)(6,14,18,43)(7,20)(9,27)(10,28)(11,29)(12,30)(13,36,39,47)(15,41,46,66)(16,25,17,42)(19,40,38,63)(21,48)(22,26)(23,49)(24,50)(31,60,45,53)(32,54)(33,56)(34,57)(35,58)(37,62,59,65)(44,64,61,68)(51,67)(52,55);(2,4)(5,9,11,34,12,35,10,33)(6,15,17,31,18,46,16,45)(7,21,23,51,24,52,22,32)(8,27,29,57,30,58,28,56)(13,37,38,61,39,59,19,44)(14,41,42,60,43,66,25,53)(20,48,49,67,50,55,26,54)(36,62,63,68,47,65,40,64)Polycyclic presentation:
2^9|a^2=eg,b^2=i,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341667);