Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,26)(15,22,29,36)(16,21,28,37)(17,23,34,42)(18,24,19,25)(27,38,31,44)(30,35,43,39)(32,40,33,41);(5,7)(6,9)(8,11)(10,12)(13,15,19,28,20,29,18,16)(14,21,24,36,26,37,25,22)(17,31,32,30,34,27,33,43)(23,39,41,38,42,35,40,44);(2,4)(7,11)(9,12)(13,16,18,29,20,28,19,15)(14,22,25,37,26,36,24,21)(17,30,33,31,34,43,32,27)(23,38,40,39,42,44,41,35);(7,11)(9,12)(13,17)(14,23)(15,27)(16,30)(18,33)(19,32)(20,34)(21,35)(22,38)(24,41)(25,40)(26,42)(28,43)(29,31)(36,44)(37,39);(6,10)(9,12)(13,18,20,19)(14,24,26,25)(15,28,29,16)(17,32,34,33)(21,22,37,36)(23,40,42,41)(27,30,31,43)(35,44,39,38)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,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, 6378115);