Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13400 |
| 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:
(1,2,6,11,7,12,30,13)(3,14,18,38,19,45,17,44)(4,20,22,37,24,42,23,9)(5,25,28,10,29,40,51,41)(8,26,34,16,33,53,32,54)(15,46,49,43,50,64,48,62)(21,39,58,60,27,36,59,56)(31,52,55,61,35,47,63,57);(1,3)(2,8)(4,15)(5,26)(6,17)(7,19)(9,35)(10,38)(11,32)(12,33)(13,34)(14,40)(16,51)(18,30)(20,55)(21,57)(22,48)(23,49)(24,50)(25,45)(27,61)(28,54)(29,53)(31,37)(36,46)(39,64)(41,44)(42,63)(43,60)(47,58)(52,59)(56,62);(1,4)(2,9)(3,15)(5,27)(6,23)(7,24)(8,31)(10,39)(11,42)(12,37)(13,20)(14,43)(16,52)(17,49)(18,48)(19,50)(21,29)(22,30)(25,60)(26,61)(28,58)(32,55)(33,35)(34,63)(36,41)(38,46)(40,56)(44,64)(45,62)(47,54)(51,59)(53,57)Polycyclic presentation:
2^9|a^2=g,d^2=h,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=dhi,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279962, #72280321, #72307196, … |
| maximal normal subgroup of | #72279962, #72280321, #72307196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54853);