Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| 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 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10)(11,22)(12,27)(13,19)(14,20)(15,36)(16,37)(17,25)(18,31)(21,28)(23,47)(24,41)(26,53)(29,55)(30,56)(32,62)(33,39)(34,44)(35,64)(38,66)(40,63)(42,60)(43,61)(45,51)(46,48)(49,69)(50,54)(52,65)(57,58)(59,70)(67,72)(68,71);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,23)(13,33)(14,25)(15,26)(16,24)(18,38)(19,41)(20,22)(21,40)(27,54)(28,48)(29,59)(30,50)(31,51)(32,49)(34,63)(35,52)(36,65)(37,39)(42,68)(43,45)(44,46)(47,56)(53,64)(55,69)(57,67)(58,71)(60,72)(61,66)(62,70);(9,12,28,58,16,32,35,61,14,30,34,60,13,29,15,31)(10,18,36,55,19,42,44,56,20,43,64,62,37,57,21,27)(11,23,48,71,24,49,52,66,25,50,63,72,33,59,26,51)(17,38,65,69,41,68,46,47,22,45,53,70,39,67,40,54);(2,5)(6,8)(9,13,14,16)(10,19,20,37)(11,24,25,33)(12,29,30,32)(15,34,35,28)(17,39,22,41)(18,42,43,57)(21,36,44,64)(23,49,50,59)(26,48,52,63)(27,55,56,62)(31,60,61,58)(38,67,45,68)(40,53,46,65)(47,69,54,70)(51,71,66,72);(1,3,4,7)(2,6,5,8)(10,20)(11,25)(18,43)(19,37)(21,44)(23,50)(24,33)(26,52)(27,56)(36,64)(42,57)(48,63)(49,59)(51,66)(55,62)(71,72)Polycyclic presentation:
2^9|c^2=gh,d^2=i,e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532193);