Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454093 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16249 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2125 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)(3,14)(4,10)(5,11)(6,29)(7,12)(8,13)(9,17)(15,33)(16,48)(18,40)(19,32)(20,53)(21,54)(22,41)(23,35)(24,36)(25,37)(26,42)(27,38)(28,39)(30,46)(31,50)(34,56)(43,47)(44,51)(45,60)(49,62)(52,59)(55,61)(57,64)(58,63);(1,3)(2,9)(4,20)(5,16)(6,19)(7,17)(8,18)(10,33)(11,31)(12,40)(13,32)(14,29)(15,23)(21,49)(22,47)(24,46)(25,45)(26,52)(27,50)(28,51)(30,35)(34,60)(36,43)(37,62)(38,44)(39,59)(41,53)(42,48)(54,58)(55,63)(56,64)(57,61);(1,4)(2,10)(3,15)(5,25)(6,23)(7,22)(8,24)(9,30)(11,37)(12,41)(13,36)(14,43)(16,49)(17,20)(18,47)(19,46)(21,26)(27,56)(28,55)(29,35)(31,58)(32,53)(33,40)(34,38)(39,61)(42,54)(44,62)(45,50)(48,57)(51,64)(52,63)(59,60);(1,5,8,28)(2,11,13,39)(3,16,18,51)(4,21,24,56)(6,27,7,26)(9,31,32,59)(10,34,36,54)(12,42,29,38)(14,44,40,48)(15,45,47,63)(17,52,19,50)(20,49,46,64)(22,55,23,25)(30,57,53,62)(33,60,43,58)(35,61,41,37)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72318063, #72318509, #72318589, … |
| maximal normal subgroup of | #72318063, #72318509, #72318589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361290);