Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,24)(15,25,30,42)(16,21,31,39)(17,22,32,41)(18,34,26,23)(20,35,38,29)(27,40,43,37)(28,36,33,44);(7,8)(10,11)(13,15,26,35,19,30,18,29)(14,20,34,42,24,38,23,25)(16,27,17,33,31,43,32,28)(21,36,41,37,39,44,22,40);(2,4)(5,7,9,8)(6,10,12,11)(13,16)(14,21)(15,27)(17,26)(18,32)(19,31)(20,36)(22,23)(24,39)(25,40)(28,29)(30,43)(33,35)(34,41)(37,42)(38,44);(5,8,9,7)(6,11,12,10)(13,17,19,32)(14,22,24,41)(15,28,30,33)(16,26,31,18)(20,37,38,40)(21,23,39,34)(25,36,42,44)(27,29,43,35);(6,12)(10,11)(16,31)(17,32)(21,39)(22,41)(27,43)(28,33)(36,44)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394998);