Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,16,11,17,10,18)(7,19,22,13,23,43,32,36)(8,14,27,38,26,39,25,40)(9,28,33,42,31,44,30,15)(20,47,49,34,50,45,55,61)(21,35,52,58,53,63,54,46)(24,41,57,64,56,65,29,37)(48,60,51,66,67,68,59,62);(1,2,3,4)(5,7,12,22,11,23,10,32)(6,13,16,43,17,36,18,19)(8,20,27,49,26,50,25,55)(9,21,33,52,31,53,30,54)(14,34,38,45,39,61,40,47)(15,35,28,58,42,63,44,46)(24,48,57,51,56,67,29,59)(37,60,41,66,64,68,65,62);(2,4)(5,8,11,26)(6,14,17,39)(7,20,23,50)(9,29,31,57)(10,27,12,25)(13,34,36,61)(15,41,42,65)(16,40,18,38)(19,45,43,47)(21,51,53,59)(22,55,32,49)(24,33,56,30)(28,37,44,64)(35,62,63,66)(46,60,58,68)(48,54,67,52);(5,9,10,30,11,31,12,33)(6,15,18,44,17,42,16,28)(7,21,32,54,23,53,22,52)(8,24,25,29,26,56,27,57)(13,35,19,46,36,63,43,58)(14,37,40,65,39,64,38,41)(20,48,55,59,50,67,49,51)(34,60,47,62,61,68,45,66)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,c^2=i,d^2=efi,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,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380668);