Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6)(7,18)(8,23)(9,28)(10,32)(11,16)(12,17)(13,21)(14,26)(15,31)(19,45)(20,49)(22,37)(24,53)(25,55)(27,41)(29,43)(30,60)(33,38)(34,42)(35,48)(36,61)(39,47)(40,62)(44,51)(46,63)(50,58)(52,57)(54,64)(56,59)(65,67)(66,68);(5,7,10,21,12,22,11,33)(6,13,16,18,17,38,32,37)(8,24,26,19,27,47,34,48)(9,20,29,50,31,51,30,61)(14,39,23,45,42,53,41,35)(15,36,43,44,28,58,60,49)(25,54,56,46,57,66,62,67)(40,64,52,65,59,68,55,63);(1,2,3,4)(5,8,10,26,12,27,11,34)(6,14,16,23,17,42,32,41)(7,19,21,47,22,48,33,24)(9,25,29,56,31,57,30,62)(13,35,18,39,38,45,37,53)(15,40,43,52,28,59,60,55)(20,46,50,66,51,67,61,54)(36,63,44,64,58,65,49,68);(5,9,10,29,12,31,11,30)(6,15,16,43,17,28,32,60)(7,20,21,50,22,51,33,61)(8,25,26,56,27,57,34,62)(13,36,18,44,38,58,37,49)(14,40,23,52,42,59,41,55)(19,46,47,66,48,67,24,54)(35,63,39,64,45,65,53,68)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305646);