Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1766 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)(13,14,19,31,18,21,17,23)(15,24,30,20,27,37,26,33)(16,22,32,42,28,35,29,36)(25,38,41,34,39,44,40,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,27)(14,20,21,33)(16,25,28,39)(17,30,19,26)(22,34,35,43)(23,37,31,24)(29,41,32,40)(36,44,42,38);(1,2,3,4)(6,9)(7,11)(14,21)(20,33)(22,35)(23,31)(24,37)(34,43)(36,42)(38,44);(5,7,8,11)(6,10,9,12)(13,16,18,28)(14,22,21,35)(15,25,27,39)(17,29,19,32)(20,34,33,43)(23,36,31,42)(24,38,37,44)(26,40,30,41);(6,9)(10,12)(16,28)(22,35)(25,39)(29,32)(34,43)(36,42)(38,44)(40,41)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427157);