Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,22)(15,23,24,19)(16,20,27,34)(17,30,31,21)(25,35,28,32)(26,33,38,37)(29,42,40,36)(39,43,41,44);(1,2,3,4)(15,24)(16,27)(19,23)(20,34)(26,38)(29,40)(33,37)(36,42);(6,9)(10,11)(13,15)(14,19)(16,28)(17,26)(18,24)(20,35)(21,33)(22,23)(25,27)(29,41)(30,37)(31,38)(32,34)(36,44)(39,40)(42,43);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,25)(17,29)(18,27)(19,32)(21,36)(22,34)(23,35)(24,28)(26,39)(30,42)(31,40)(33,43)(37,44)(38,41);(5,7)(6,10)(8,12)(9,11)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,35)(26,37)(27,34)(28,32)(29,36)(30,31)(33,38)(39,44)(40,42)(41,43);(13,17,18,31)(14,21,22,30)(15,26,24,38)(16,29,27,40)(19,33,23,37)(20,36,34,42)(25,39,28,41)(32,43,35,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622625);