Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395244 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14722 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23434 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1672 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)(4,19)(5,25)(6,29)(7,12)(8,13)(9,17)(10,23)(11,28)(15,31)(16,44)(18,33)(20,35)(21,49)(22,38)(24,37)(26,39)(27,55)(30,34)(32,56)(36,50)(40,46)(41,61)(42,58)(43,59)(45,53)(47,52)(48,64)(51,54)(57,63)(60,62);(1,3,6,17,8,18,7,30)(2,9,12,14,13,34,29,33)(4,20,22,15,24,42,23,43)(5,16,26,45,28,46,27,56)(10,35,37,59,38,58,19,31)(11,32,39,40,25,53,55,44)(21,48,50,41,52,62,51,63)(36,60,49,61,54,64,47,57);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,31)(11,36)(12,37)(13,38)(14,35)(16,41)(17,42)(18,43)(19,29)(20,30)(25,54)(26,50)(27,51)(28,52)(32,57)(33,58)(34,59)(39,49)(40,60)(44,64)(45,62)(46,63)(47,55)(48,56)(53,61);(1,5,6,26,8,28,7,27)(2,11,12,39,13,25,29,55)(3,16,17,45,18,46,30,56)(4,21,22,50,24,52,23,51)(9,32,14,40,34,53,33,44)(10,36,37,49,38,54,19,47)(15,41,42,62,43,63,20,48)(31,57,35,60,59,61,58,64)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302440);