Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622982 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #33604 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20)(11,21,22,43)(12,17,27,38)(13,30,32,58)(14,33,31,19)(16,34,35,25)(18,41,42,23)(24,44,47,65)(26,53,48,46)(28,56,54,71)(29,57,55,40)(36,59,61,51)(37,49,45,60)(39,64,63,50)(52,70,68,67)(62,69,66,72);(2,5)(6,8)(11,22)(16,35)(21,43)(24,47)(25,34)(26,48)(36,61)(37,45)(44,65)(46,53)(49,60)(51,59)(52,68)(62,66)(67,70)(69,72);(9,11,23,49,14,26,32,34,15,22,41,60,31,48,13,25)(10,16,30,46,19,37,42,43,20,35,58,53,33,45,18,21)(12,24,50,69,29,52,54,59,27,47,64,72,55,68,28,51)(17,36,56,67,40,62,63,65,38,61,71,70,57,66,39,44);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,24)(13,28)(14,29)(15,27)(16,36)(18,39)(19,40)(20,38)(21,44)(22,47)(23,50)(25,51)(26,52)(30,56)(31,55)(32,54)(33,57)(34,59)(35,61)(37,62)(41,64)(42,63)(43,65)(45,66)(46,67)(48,68)(49,69)(53,70)(58,71)(60,72);(12,27)(17,38)(24,47)(28,54)(29,55)(36,61)(39,63)(40,57)(44,65)(50,64)(51,59)(52,68)(56,71)(62,66)(67,70)(69,72)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,c^b=ci,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530461);