Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2023 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11384 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,11,27,13)(3,14,17,45,19,44,18,38)(4,20,23,37,22,9,33,36)(5,24,28,10,29,41,51,40)(8,25,32,54,31,53,42,16)(15,46,49,63,48,43,50,64)(21,55,57,39,26,59,60,35)(30,56,61,52,34,58,62,47);(1,3)(2,8,11,31)(4,15)(5,25)(6,17)(7,19)(9,34,20,30)(10,38,40,45)(12,32,13,42)(14,24,44,41)(16,51)(18,27)(21,56)(22,48)(23,49)(26,58)(28,54)(29,53)(33,50)(35,63,39,64)(36,62,37,61)(43,59,46,55)(47,60)(52,57);(1,4)(2,9,11,20)(3,15)(5,26)(6,23)(7,22)(8,30,31,34)(10,39,40,35)(12,36,13,37)(14,43,44,46)(16,52)(17,49)(18,50)(19,48)(21,29)(24,55,41,59)(25,58)(27,33)(28,60)(32,61,42,62)(38,63,45,64)(47,54)(51,57)(53,56)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=hi,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=dhi,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279896, #72279962, #72307177, … |
| maximal normal subgroup of | #72279896, #72279962, #72307177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56669);