Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6)(7,19)(8,14,28,42)(9,30)(10,17)(11,18)(12,16)(13,22)(15,34)(20,47,25,53)(21,36)(23,37)(24,38)(26,55,58,66)(27,43,29,41)(31,61)(32,46)(33,45)(35,50,39,49)(40,57,67,59)(44,52)(48,63,54,64)(51,60)(56,68,62,65);(2,3)(5,7,11,23)(6,13,18,38)(8,25,28,20)(9,31,33,52)(10,24,12,22)(14,39,42,35)(15,44,46,61)(16,37,17,19)(21,34,51,32)(26,56,58,62)(27,49,29,50)(30,60,45,36)(40,65,67,68)(41,47,43,53)(48,57,54,59)(55,63,66,64);(1,2,4,3)(5,8,12,29,11,28,10,27)(6,14,16,41,18,42,17,43)(7,20,24,50,23,25,22,49)(9,26,34,59,33,58,32,57)(13,35,19,47,38,39,37,53)(15,40,45,66,46,67,30,55)(21,48,52,56,51,54,31,62)(36,63,44,68,60,64,61,65);(1,3,4,2)(5,9,12,34,11,33,10,32)(6,15,17,30,18,46,16,45)(7,21,24,52,23,51,22,31)(8,26,29,59,28,58,27,57)(13,36,37,61,38,60,19,44)(14,40,43,55,42,67,41,66)(20,48,50,56,25,54,49,62)(35,63,53,65,39,64,47,68)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ef,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 341625);