Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14796 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,14)(3,15,18,31,19,32,20,9)(4,10,22,35,23,36,28,39)(5,24,25,11,26,37,27,38)(16,42,44,29,45,40,50,33)(17,46,47,41,49,30,48,55)(21,51,52,59,53,56,54,34)(43,58,62,60,63,57,64,61);(1,3)(2,9)(4,16)(5,17)(6,20)(7,18)(8,19)(10,33)(11,30)(12,32)(13,31)(14,15)(21,43)(22,50)(23,45)(24,55)(25,48)(26,49)(27,47)(28,44)(29,36)(34,58)(35,40)(37,41)(38,46)(39,42)(51,61)(52,64)(53,63)(54,62)(56,60)(57,59);(1,4)(2,10)(3,16)(5,21)(6,22)(7,28)(8,23)(9,29)(11,34)(12,35)(13,36)(14,39)(15,40)(17,43)(18,44)(19,45)(20,50)(24,56)(25,52)(26,53)(27,54)(30,57)(31,33)(32,42)(37,51)(38,59)(41,60)(46,58)(47,62)(48,64)(49,63)(55,61);(1,5,6,27,8,26,7,25)(2,11,12,24,13,38,14,37)(3,17,20,47,19,49,18,48)(4,21,22,54,23,53,28,52)(9,30,32,55,31,46,15,41)(10,34,35,56,36,59,39,51)(16,43,50,62,45,63,44,64)(29,57,42,61,33,58,40,60)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285493, #72285582, C2×#400132, … |
| maximal normal subgroup of | #72285493, #72285582, C2×#400132, … |
| maximal quotient of | C2×#400132 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307328);