Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| 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 | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,5,3,4,6,8,7)(9,10)(11,21)(12,17)(13,29)(14,34)(15,18)(16,23)(19,32)(20,31)(22,46)(24,53)(25,47)(26,41)(27,56)(28,59)(30,44)(33,45)(35,43)(36,49)(37,52)(38,61)(39,58)(40,55)(42,51)(48,68)(50,71)(54,67)(57,66)(60,64)(62,72)(63,69)(65,70);(2,6)(3,7)(9,11,15,25)(10,16,18,37)(12,22,26,48)(13,30,33,61)(14,35,31,24)(17,36,41,63)(19,43,42,53)(20,44,34,38)(21,45,47,29)(23,51,52,32)(27,57,54,72)(28,60,55,50)(39,64,65,71)(40,66,59,62)(46,67,68,56)(49,70,69,58);(1,3,5,6,4,7,8,2)(9,12,15,26)(10,17,18,41)(11,22,25,48)(13,27,33,54)(14,28,31,55)(16,36,37,63)(19,39,42,65)(20,40,34,59)(21,46,47,68)(23,49,52,69)(24,50,35,60)(29,56,45,67)(30,57,61,72)(32,58,51,70)(38,62,44,66)(43,64,53,71);(10,18)(12,26)(16,37)(19,42)(20,34)(21,47)(22,48)(27,54)(28,55)(29,45)(38,44)(43,53)(49,69)(50,60)(57,72)(58,70)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=eg,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391026);