Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14171 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,19,33,28,32)(4,10,22,37,23,38,29,36)(5,24,25,11,26,40,27,39)(16,42,45,56,44,34,20,30)(17,31,47,57,48,58,46,41)(21,50,53,60,52,54,51,35)(43,55,62,61,63,64,49,59);(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)(25,47)(26,48)(27,46)(29,44)(30,37)(35,59)(36,56)(38,42)(39,58)(40,57)(43,53)(50,55)(51,63)(52,62)(54,64)(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)(24,54)(25,53)(26,52)(27,51)(31,55)(32,56)(33,42)(39,60)(40,50)(41,59)(46,49)(47,62)(48,63)(57,61)(58,64);(1,5,7,25,8,26,6,27)(2,11,12,40,14,39,13,24)(3,17,18,47,19,48,28,46)(4,21,22,53,23,52,29,51)(9,31,33,57,32,58,15,41)(10,35,37,50,38,60,36,54)(16,43,45,62,44,63,20,49)(30,55,42,61,56,64,34,59)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282958);