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 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,28)(13,23)(14,21)(15,37)(16,20)(17,26)(18,32)(22,29)(24,49)(25,39)(27,55)(30,59)(31,58)(33,57)(34,45)(35,48)(36,47)(38,66)(40,64)(41,53)(42,60)(43,63)(44,62)(46,50)(51,71)(52,56)(54,65)(61,69)(67,70)(68,72);(1,3)(2,5)(4,8)(6,7)(9,11,14,26)(10,17,21,19)(12,24,31,52)(13,34,16,25)(15,27,36,54)(18,38,43,41)(20,45,23,39)(22,40,48,46)(28,56,58,49)(29,50,35,64)(30,61,33,51)(32,53,63,66)(37,65,47,55)(42,68,44,67)(57,69,59,71)(60,70,62,72);(9,12,29,60,16,33,36,63,14,31,35,62,13,30,15,32)(10,18,37,59,23,44,48,58,21,43,47,57,20,42,22,28)(11,24,50,70,25,51,54,66,26,52,64,72,34,61,27,53)(17,38,65,71,39,67,46,49,19,41,55,69,45,68,40,56);(3,8)(5,7)(9,11)(10,19)(12,24)(13,34)(14,26)(15,27)(16,25)(17,21)(18,41)(20,39)(22,46)(23,45)(28,49)(29,50)(30,61)(31,52)(32,53)(33,51)(35,64)(36,54)(37,55)(38,43)(40,48)(42,67)(44,68)(47,65)(56,58)(57,71)(59,69)(60,70)(62,72)(63,66);(2,6)(5,7)(9,13,14,16)(10,20,21,23)(11,25,26,34)(12,30,31,33)(15,35,36,29)(17,39,19,45)(18,42,43,44)(22,47,48,37)(24,51,52,61)(27,50,54,64)(28,57,58,59)(32,62,63,60)(38,67,41,68)(40,65,46,55)(49,69,56,71)(53,70,66,72)Polycyclic presentation:
2^9|b^2=i,c^2=gh,e^2=i,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=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532055);