Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14722 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1691 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)(4,20)(5,24)(6,13)(7,12)(8,14)(9,18)(10,22)(11,26)(16,30)(17,46)(19,37)(21,51)(23,38)(25,36)(27,32)(28,35)(29,39)(31,48)(33,45)(34,52)(40,44)(41,49)(42,57)(43,50)(47,55)(53,56)(54,59)(58,62)(60,63)(61,64);(1,3,6,18,8,19,7,27)(2,9,12,15,14,32,13,37)(4,16,22,43,23,45,28,44)(5,17,25,47,26,49,29,48)(10,33,20,50,35,30,38,40)(11,31,36,41,24,55,39,46)(21,42,52,61,54,62,53,63)(34,58,51,64,56,57,59,60);(1,4)(2,10)(3,16)(5,21)(6,22)(7,28)(8,23)(9,30)(11,34)(12,20)(13,38)(14,35)(15,40)(17,42)(18,43)(19,45)(24,56)(25,52)(26,54)(27,44)(29,53)(31,57)(32,33)(36,51)(37,50)(39,59)(41,60)(46,64)(47,61)(48,63)(49,62)(55,58);(1,5,6,25,8,26,7,29)(2,11,12,36,14,24,13,39)(3,17,18,47,19,49,27,48)(4,21,22,52,23,54,28,53)(9,31,15,41,32,55,37,46)(10,34,20,51,35,56,38,59)(16,42,43,61,45,62,44,63)(30,57,40,60,33,58,50,64)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302388);