Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | #3455 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10,13,19,14,20,28,40,15,18,29,38,30,39,50,60)(11,16,23,32,24,33,46,54,22,34,44,55,45,56,64,69)(12,17,26,36,27,37,49,59,25,35,47,57,48,58,65,70)(21,31,42,52,43,53,63,68,41,51,61,66,62,67,71,72);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,31)(18,34)(19,32)(20,33)(25,41)(26,42)(27,43)(28,46)(29,44)(30,45)(35,51)(36,52)(37,53)(38,55)(39,56)(40,54)(47,61)(48,62)(49,63)(50,64)(57,66)(58,67)(59,68)(60,69)(65,71)(70,72);(2,5)(6,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,31)(18,35)(19,36)(20,37)(22,41)(23,42)(24,43)(28,49)(29,47)(30,48)(32,52)(33,53)(34,51)(38,57)(39,58)(40,59)(44,61)(45,62)(46,63)(50,65)(54,68)(55,66)(56,67)(60,70)(64,71)(69,72);(1,3,4,7)(2,6,5,8)(12,25)(17,35)(21,41)(26,47)(27,48)(31,51)(36,57)(37,58)(42,61)(43,62)(49,65)(52,66)(53,67)(59,70)(63,71)(68,72);(10,18)(11,22)(17,35)(19,38)(20,39)(21,41)(23,44)(24,45)(36,57)(37,58)(40,60)(42,61)(43,62)(46,64)(59,70)(63,71)Polycyclic presentation:
2^9|a^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf,e^b=ei,d^c=di152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529918);