Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1879 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,33,43,9,15,5,29,32,46,6,13)(3,16,21,39,22,38,20,37,18,44,19,47,34,60,30,10)(4,23,26,42,27,45,25,48,28,59,24,11,35,40,31,41)(17,50,54,49,55,61,56,64,51,63,52,62,57,36,53,58);(1,3)(2,10,15,37)(4,17)(5,30)(6,19)(7,20)(8,34)(9,18)(11,36,42,61)(12,38,29,60)(13,44,43,16)(14,47,46,39)(21,33)(22,32)(23,58,59,64)(24,53)(25,54)(26,56)(27,57)(28,51)(31,52)(35,55)(40,49,45,62)(41,63,48,50);(1,4)(2,11)(3,17)(5,31)(6,24)(7,25)(8,35)(9,28)(10,36)(12,23)(13,45)(14,48)(15,42)(16,49)(18,51)(19,53)(20,54)(21,56)(22,57)(26,33)(27,32)(29,59)(30,52)(34,55)(37,61)(38,58)(39,63)(40,43)(41,46)(44,62)(47,50)(60,64);(3,18)(10,37)(16,44)(17,51)(19,21)(20,30)(22,34)(36,61)(38,60)(39,47)(49,62)(50,63)(52,54)(53,56)(55,57)(58,64)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279896, #72280077, #72280165, … |
| maximal normal subgroup of | #72279896, #72280077, #72280165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401735);