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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2044 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)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,31,40)(16,32,33,42)(17,23,36,43)(18,25,19,24)(21,30,41,29)(22,34,39,35)(28,38,37,44);(5,7)(6,9)(8,11)(10,12)(13,15,19,30,20,31,18,29)(14,21,25,40,26,41,24,27)(16,28,35,17,33,37,34,36)(22,38,32,43,39,44,42,23);(2,4)(7,11)(9,12)(13,16,18,34,20,33,19,35)(14,22,24,42,26,39,25,32)(15,28,29,36,31,37,30,17)(21,38,27,23,41,44,40,43);(7,11)(9,12)(13,17)(14,23)(15,16)(18,28)(19,37)(20,36)(21,39)(22,41)(24,44)(25,38)(26,43)(27,32)(29,34)(30,35)(31,33)(40,42);(6,10)(9,12)(16,33)(17,36)(22,39)(23,43)(28,37)(32,42)(34,35)(38,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378099);