Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
2231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,26)(16,22)(17,33)(18,24)(19,25)(20,23)(21,29)(27,44)(28,37)(30,43)(31,42)(32,38)(34,40)(35,41)(36,39);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,30)(17,28)(18,35)(19,29)(20,36)(22,40)(23,37)(24,42)(25,26)(27,31)(32,34)(33,39)(38,43)(41,44);(1,2,3,4)(13,16)(14,22)(15,27)(17,31)(18,29)(19,32)(20,35)(21,24)(23,41)(25,38)(26,44)(28,34)(30,36)(33,42)(37,40)(39,43);(6,10)(9,11)(13,17,19,20)(14,23,25,33)(15,28,29,36)(16,31,32,35)(18,30,27,34)(21,37,26,39)(22,41,38,42)(24,40,44,43);(7,12)(9,11)(13,18,19,27)(14,24,25,44)(15,16,29,32)(17,34,20,30)(21,38,26,22)(23,43,33,40)(28,35,36,31)(37,41,39,42)Polycyclic presentation:
2^9|c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589412);