Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13396 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10969 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,8,13,7,30)(3,14,18,43,19,42,17,39)(4,20,22,37,23,38,29,10)(5,24,26,40,28,57,27,11)(9,25,34,50,33,49,32,16)(15,44,47,41,48,63,46,61)(21,36,54,52,55,62,58,56)(31,59,60,53,35,45,51,64);(1,3)(2,9)(4,15)(5,25)(6,18)(7,17)(8,19)(10,35)(11,39)(12,34)(13,33)(14,24)(16,27)(20,51)(21,53)(22,47)(23,48)(26,50)(28,49)(29,46)(30,32)(31,37)(36,44)(38,60)(40,43)(41,52)(42,57)(45,54)(55,64)(56,61)(58,59)(62,63);(1,4)(2,10)(3,15)(5,21)(6,29)(7,22)(8,23)(9,31)(11,36)(12,38)(13,37)(14,41)(16,45)(17,47)(18,46)(19,48)(20,30)(24,56)(25,53)(26,58)(27,54)(28,55)(32,60)(33,35)(34,51)(39,63)(40,62)(42,61)(43,44)(49,64)(50,59)(52,57)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279962, #72280557, #72307214, … |
| maximal normal subgroup of | #72279962, #72280557, #72307214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54856);