Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417626 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15257 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,9,19,34)(4,20,23,40)(5,25,28,38)(6,13,7,12)(10,29,37,24)(11,27,39,26)(15,41,43,35)(16,45,47,59)(17,33,18,32)(21,50,44,30)(22,36,53,62)(31,48,58,46)(42,57,63,64)(49,54,56,55)(51,61,52,60);(1,3)(2,9)(4,21)(5,16)(6,17)(7,18)(8,19)(10,35)(11,31)(12,32)(13,33)(14,34)(15,24)(20,30)(22,52)(23,44)(25,45)(26,46)(27,48)(28,47)(29,43)(36,61)(37,41)(38,59)(39,58)(40,50)(42,55)(49,64)(51,53)(54,63)(56,57)(60,62);(1,4)(2,10)(3,15)(5,22)(6,29)(7,24)(8,23)(9,30)(11,36)(12,40)(13,20)(14,37)(16,42)(17,21)(18,44)(19,43)(25,56)(26,55)(27,54)(28,53)(31,57)(32,35)(33,41)(34,50)(38,49)(39,62)(45,60)(46,51)(47,63)(48,52)(58,64)(59,61);(1,5)(2,11)(3,16)(4,22)(6,26)(7,27)(8,28)(9,31)(10,36)(12,25)(13,38)(14,39)(15,42)(17,46)(18,48)(19,47)(20,49)(21,51)(23,53)(24,54)(29,55)(30,57)(32,45)(33,59)(34,58)(35,60)(37,62)(40,56)(41,61)(43,63)(44,52)(50,64)Polycyclic presentation:
2^9|a^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72264928, #72264934, #72265017, … |
| maximal normal subgroup of | #72264928, #72264934, #72265017, … |
| maximal quotient of | #72313941 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324823);