Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219498 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,17,10,34)(7,18,22,38,23,37,21,13)(8,24,27,14,29,41,28,42)(9,30,32,43,33,60,31,15)(19,35,25,54,46,52,51,39)(20,47,49,62,50,36,48,44)(26,56,57,53,59,40,58,64)(45,63,55,61,65,68,67,66);(1,2,3,4)(5,7)(6,13,17,38)(8,25,29,51)(9,20)(10,21)(11,22)(12,23)(14,39)(15,36,43,47)(16,18,34,37)(19,27,46,28)(24,52)(26,55,59,67)(30,44,60,62)(31,48)(32,49)(33,50)(35,41)(40,63)(42,54)(45,57,65,58)(53,66)(56,68)(61,64);(2,4)(5,8)(6,14,17,42)(7,19,23,46)(9,26)(10,27)(11,28)(12,29)(13,35)(15,40,43,56)(16,24,34,41)(18,39)(20,45,50,65)(21,25,22,51)(30,53,60,64)(31,57)(32,58)(33,59)(36,61)(37,54)(38,52)(44,63)(47,66)(48,55,49,67)(62,68);(5,9,11,32,12,33,10,31)(6,15,34,60,17,43,16,30)(7,20,22,49,23,50,21,48)(8,26,28,58,29,59,27,57)(13,36,37,62,38,47,18,44)(14,40,41,64,42,56,24,53)(19,45,51,67,46,65,25,55)(35,61,54,68,52,66,39,63)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274575);