Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,4,6)(3,7,8,5)(9,10,13,19)(11,18,23,16)(12,26,27,62)(14,32,30,61)(15,35,31,21)(17,40,42,28)(20,47,46,34)(22,50,51,60)(24,56,54,65)(25,39,55,45)(29,63,36,48)(33,49,64,43)(37,52,41,69)(38,58,44,70)(53,67,59,71)(57,66,72,68);(1,3)(2,5)(4,8)(6,7)(9,11,13,23)(10,16,19,18)(12,22,27,51)(14,24,30,54)(15,25,31,55)(17,37,42,41)(20,38,46,44)(21,39,35,45)(26,60,62,50)(28,52,40,69)(29,53,36,59)(32,65,61,56)(33,57,64,72)(34,58,47,70)(43,66,49,68)(48,67,63,71);(9,12,13,27)(10,17,19,42)(11,22,23,51)(14,33,30,64)(15,36,31,29)(16,37,18,41)(20,48,46,63)(21,49,35,43)(24,57,54,72)(25,59,55,53)(26,61,62,32)(28,47,40,34)(38,67,44,71)(39,68,45,66)(50,65,60,56)(52,70,69,58);(3,8)(5,7)(9,11)(10,18)(12,22)(13,23)(14,24)(15,25)(16,19)(17,41)(20,44)(21,45)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,39)(36,59)(37,42)(38,46)(40,69)(43,68)(47,70)(48,71)(49,66)(60,62)(61,65)(63,67)(64,72);(11,23)(16,18)(22,51)(24,54)(25,55)(37,41)(38,44)(39,45)(50,60)(52,69)(53,59)(56,65)(57,72)(58,70)(66,68)(67,71)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530959);