Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396100 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2414 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15123 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1801 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)(3,15)(4,10)(5,24)(6,13)(7,12)(8,14)(9,18)(11,26)(16,42)(17,46)(19,33)(20,30)(21,50)(22,37)(23,38)(25,40)(27,39)(28,32)(29,36)(31,48)(34,44)(35,51)(41,47)(43,55)(45,56)(49,58)(52,59)(53,57)(54,60)(61,63)(62,64);(1,3)(2,9)(4,20)(5,17)(6,28)(7,18)(8,19)(10,34)(11,31)(12,33)(13,15)(14,32)(16,22)(21,49)(23,45)(24,41)(25,53)(26,47)(27,48)(29,44)(30,37)(35,58)(36,56)(38,42)(39,57)(40,46)(43,51)(50,64)(52,63)(54,62)(55,59)(60,61);(1,4)(2,10)(3,16)(5,21)(6,29)(7,22)(8,23)(9,30)(11,35)(12,37)(13,36)(14,38)(15,34)(17,43)(18,45)(19,44)(20,28)(24,50)(25,54)(26,51)(27,52)(31,55)(32,56)(33,42)(39,59)(40,60)(41,58)(46,64)(47,63)(48,62)(49,53)(57,61);(1,5,7,26,8,27,6,25)(2,11,13,24,14,40,12,39)(3,17,18,47,19,48,28,53)(4,21,22,51,23,52,29,54)(9,31,15,41,32,46,33,57)(10,35,36,50,38,60,37,59)(16,43,45,63,44,62,20,49)(30,55,34,58,56,64,42,61)Polycyclic presentation:
2^9|d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318923);