Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| 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 | |
1278 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)(9,10,13,19)(11,18,23,16)(12,26,28,55)(14,33,31,61)(15,34,32,21)(17,38,40,29)(20,45,44,27)(22,46,48,54)(24,53,51,62)(25,37,52,43)(30,60,59,57)(35,49,39,64)(36,47,42,67)(41,58,56,65)(50,71,70,68)(63,72,66,69);(1,3)(2,5)(4,8)(6,7)(9,11,13,23)(10,16,19,18)(12,22,28,48)(14,24,31,51)(15,25,32,52)(17,35,40,39)(20,36,44,42)(21,37,34,43)(26,54,55,46)(27,47,45,67)(29,49,38,64)(30,50,59,70)(33,62,61,53)(41,63,56,66)(57,71,60,68)(58,69,65,72);(9,12,27,58,15,30,31,38,13,28,45,65,32,59,14,29)(10,17,33,57,21,41,44,55,19,40,61,60,34,56,20,26)(11,22,47,69,25,50,51,64,23,48,67,72,52,70,24,49)(16,35,62,71,37,63,42,46,18,39,53,68,43,66,36,54);(3,8)(5,7)(9,11)(10,18)(12,22)(13,23)(14,24)(15,25)(16,19)(17,39)(20,42)(21,43)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,37)(35,40)(36,44)(38,64)(41,66)(45,67)(54,55)(56,63)(57,68)(58,69)(59,70)(60,71)(61,62)(65,72);(2,6)(5,7)(10,19)(11,23)(12,28)(20,44)(21,34)(24,51)(25,52)(29,38)(30,59)(33,61)(35,39)(46,54)(47,67)(58,65)(63,66)(68,71)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,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^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532087);