Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15267 |
| 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 | |
2210 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)(5,23,26,11)(6,27)(7,12)(8,13)(9,17)(15,35)(16,40,47,43)(18,33)(19,49)(20,36,53,55)(21,38)(22,39)(24,32,48,57)(25,41,28,42)(29,34)(30,37)(31,45)(44,59,51,62)(46,58)(50,64,63,56)(52,61,54,60);(1,3,6,17,8,18,7,29)(2,9,12,14,13,34,27,33)(4,19,21,15,22,45,30,46)(5,24,25,47,26,48,28,16)(10,35,37,49,39,58,38,31)(11,40,41,57,23,43,42,32)(20,51,52,63,53,44,54,50)(36,56,60,62,55,64,61,59);(1,4,8,22)(2,10,13,39)(3,15,18,46)(5,20,26,53)(6,21,7,30)(9,31,34,49)(11,36,23,55)(12,37,27,38)(14,35,33,58)(16,44,47,51)(17,45,29,19)(24,50,48,63)(25,52,28,54)(32,56,57,64)(40,62,43,59)(41,60,42,61);(1,5)(2,11)(3,16)(4,20)(6,28)(7,25)(8,26)(9,32)(10,36)(12,42)(13,23)(14,43)(15,44)(17,48)(18,47)(19,50)(21,54)(22,53)(24,29)(27,41)(30,52)(31,56)(33,40)(34,57)(35,59)(37,61)(38,60)(39,55)(45,63)(46,51)(49,64)(58,62)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72286090, #72286134, #72286176, … |
| maximal normal subgroup of | #72286090, #72286134, #72286176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326243);