Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2718 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)(7,11)(8,10)(9,12)(13,14)(15,19)(16,20)(17,22)(18,23)(21,25)(24,33)(26,34)(27,35)(28,37)(29,38)(30,39)(31,40)(32,41)(36,42)(43,44);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,27)(17,30)(18,25)(20,35)(21,23)(22,39)(24,28)(26,32)(29,43)(31,42)(33,37)(34,41)(36,40)(38,44);(2,4)(13,16,18,29)(14,20,23,38)(15,24,25,42)(17,31,32,28)(19,33,21,36)(22,40,41,37)(26,43,30,27)(34,44,39,35);(2,4)(6,10)(7,12)(13,15)(14,21)(16,24)(17,26)(18,25)(19,23)(20,36)(22,39)(27,28)(29,42)(30,32)(31,43)(33,38)(34,41)(35,40)(37,44);(15,25)(19,21)(24,42)(26,30)(27,43)(33,36)(34,39)(35,44)Polycyclic presentation:
2^9|a^2=g,c^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410347);