Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3541 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2998 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)(13,14,19,22,18,23,17,24)(15,25,27,20,29,38,28,35)(16,30,31,36,33,21,32,37)(26,40,43,44,42,39,41,34);(5,6)(7,10)(8,9)(11,12)(13,15,18,29)(14,20,23,35)(16,26,33,42)(17,27,19,28)(21,34,30,44)(22,25,24,38)(31,41,32,43)(36,40,37,39);(2,4)(6,9)(10,12)(13,16)(14,21)(15,26)(17,32)(18,33)(19,31)(20,34)(22,37)(23,30)(24,36)(25,39)(27,43)(28,41)(29,42)(35,44)(38,40);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,22,23,24)(15,27,29,28)(16,31,33,32)(20,25,35,38)(21,36,30,37)(26,41,42,43)(34,40,44,39);(7,11)(10,12)(14,23)(20,35)(21,30)(22,24)(25,38)(34,44)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,f^b=fi,d^c=di,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449831);