Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,8)(4,6)(5,7)(9,10,16,23,14,21,17,41)(11,24,29,54,27,18,30,45)(12,31,35,46,33,64,42,52)(13,20,38,51,37,22,15,39)(19,40,48,32,47,66,49,34)(25,55,58,69,57,63,62,72)(26,50,28,60,36,44,59,67)(43,68,70,56,53,61,71,65);(1,3,5,4)(2,6,7,8)(9,11,14,27)(10,18,21,24)(12,25,33,57)(13,36,37,26)(15,28,38,59)(16,29,17,30)(19,43,47,53)(20,50,22,44)(23,45,41,54)(31,63,64,55)(32,65,34,56)(35,58,42,62)(39,67,51,60)(40,61,66,68)(46,72,52,69)(48,70,49,71);(9,12)(10,19)(11,25)(13,32)(14,33)(15,40)(16,35)(17,42)(18,43)(20,46)(21,47)(22,52)(23,48)(24,53)(26,56)(27,57)(28,61)(29,58)(30,62)(31,39)(34,37)(36,65)(38,66)(41,49)(44,69)(45,70)(50,72)(51,64)(54,71)(55,60)(59,68)(63,67);(1,4,5,3)(2,6,7,8)(9,13,17,15,14,37,16,38)(10,20,41,39,21,22,23,51)(11,26,30,59,27,36,29,28)(12,32,42,40,33,34,35,66)(18,44,54,60,24,50,45,67)(19,46,49,31,47,52,48,64)(25,56,62,68,57,65,58,61)(43,69,71,55,53,72,70,63)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=eg,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403192);