Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35660 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,24,19,25,17,23)(15,20,27,22,28,35,33,39)(16,29,31,38,32,37,30,21)(26,40,41,44,42,36,43,34);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,26)(17,33)(18,27)(19,28)(21,34)(22,24)(23,39)(25,35)(29,40)(30,43)(31,41)(32,42)(36,37)(38,44);(2,4)(13,16,19,32)(14,21,25,38)(15,26,28,42)(17,31,18,30)(20,34,35,44)(22,36,39,40)(23,29,24,37)(27,43,33,41);(6,10)(7,12)(13,15)(14,22)(16,26)(17,27)(18,33)(19,28)(20,24)(21,36)(23,35)(25,39)(29,44)(30,41)(31,43)(32,42)(34,37)(38,40);(13,17,19,18)(14,23,25,24)(15,27,28,33)(16,30,32,31)(20,22,35,39)(21,37,38,29)(26,41,42,43)(34,40,44,36)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410433);