Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #441216 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15674 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1179 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)(3,14,18,9)(4,19,23,38)(5,24)(6,13)(7,29)(8,12)(10,22,37,30)(11,27)(15,42)(16,39,47,32)(17,34,28,33)(20,41)(21,50,52,48)(25,54)(26,40)(31,44)(35,45)(36,56,53,51)(43,61)(46,59,55,58)(49,62)(57,63)(60,64);(1,3,7,17,8,18,6,28)(2,9,29,33,12,14,13,34)(4,20,22,45,23,44,30,15)(5,16,26,46,27,47,25,55)(10,35,19,31,37,42,38,41)(11,39,54,59,24,32,40,58)(21,49,51,64,52,63,56,43)(36,61,48,62,53,60,50,57);(1,4)(2,10,12,37)(3,15)(5,21)(6,22)(7,30)(8,23)(9,31,14,41)(11,36,24,53)(13,19,29,38)(16,43)(17,44)(18,45)(20,28)(25,51)(26,56)(27,52)(32,57,39,62)(33,35,34,42)(40,48,54,50)(46,63)(47,64)(49,55)(58,60,59,61);(1,5,8,27)(2,11)(3,16,18,47)(4,21,23,52)(6,25,7,26)(9,32)(10,36)(12,24)(13,40)(14,39)(15,43,45,64)(17,46,28,55)(19,48)(20,49,44,63)(22,51,30,56)(29,54)(31,57)(33,58)(34,59)(35,60)(37,53)(38,50)(41,62)(42,61)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348413);