Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394328 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2617 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14716 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2553 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)(6,28)(7,12)(8,13)(9,17)(15,32)(16,40)(18,41)(19,31)(20,48)(21,50)(22,38)(23,34)(24,35)(25,54)(26,37)(27,36)(29,43)(30,45)(33,52)(39,44)(42,60)(46,56)(47,61)(49,55)(51,58)(53,59)(57,63)(62,64);(1,3,6,17,8,18,7,19)(2,9,12,14,13,31,28,41)(4,20,22,15,24,43,23,44)(5,16,25,45,26,47,27,46)(10,32,34,48,35,39,38,29)(11,30,36,40,37,56,54,61)(21,49,51,42,52,62,53,63)(33,57,58,64,50,60,59,55);(1,4)(2,10)(3,15)(5,21)(6,23)(7,22)(8,24)(9,29)(11,33)(12,38)(13,35)(14,39)(16,42)(17,20)(18,44)(19,43)(25,53)(26,52)(27,51)(28,34)(30,55)(31,48)(32,41)(36,59)(37,50)(40,60)(45,49)(46,62)(47,63)(54,58)(56,64)(57,61);(1,5)(2,11)(3,16)(4,21)(6,25)(7,27)(8,26)(9,30)(10,33)(12,36)(13,37)(14,40)(15,42)(17,45)(18,47)(19,46)(20,49)(22,51)(23,53)(24,52)(28,54)(29,55)(31,56)(32,57)(34,58)(35,50)(38,59)(39,60)(41,61)(43,62)(44,63)(48,64)Polycyclic presentation:
2^9|b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301524);