Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,25)(16,21)(17,22)(18,24)(19,23)(20,28)(26,39)(27,34)(29,36)(30,38)(31,37)(32,35)(33,40)(41,43)(42,44);(2,3)(6,10)(11,12)(13,15)(14,20)(16,26)(17,27)(18,28)(19,32)(21,33)(22,35)(23,34)(24,25)(29,41)(30,40)(31,42)(36,44)(37,43)(38,39);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,26)(17,31)(18,30)(19,29)(20,33)(22,37)(23,36)(24,38)(25,39)(27,42)(28,40)(32,41)(34,44)(35,43);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17)(14,22)(16,29)(18,19)(21,36)(23,24)(26,40)(27,32)(30,31)(33,39)(34,35)(37,38);(6,10)(11,12)(13,17,18,19)(14,23,24,22)(15,27,28,32)(16,29,30,31)(20,34,25,35)(21,37,38,36)(26,41,40,42)(33,43,39,44)Polycyclic presentation:
2^9|c^2=fgi,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468810);