Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,18,23,33,40)(15,24,27,42,28,19,34,37)(16,29,32,38,31,21,30,20)(25,39,44,35,26,36,43,41);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,25)(17,27)(18,28)(20,35)(21,39)(22,37)(23,24)(26,31)(29,36)(30,43)(32,44)(33,34)(38,41)(40,42);(5,7)(6,9)(8,12)(10,11)(13,16)(14,20)(15,25)(17,30)(18,31)(19,35)(21,22)(23,38)(24,41)(26,28)(27,43)(29,40)(32,33)(34,44)(36,42)(37,39);(1,3,4,2)(7,12)(9,11)(13,16)(14,21)(15,26)(17,32)(18,31)(19,36)(20,22)(23,29)(24,39)(25,28)(27,43)(30,33)(34,44)(35,42)(37,41)(38,40);(6,10)(9,11)(13,17,18,33)(14,22,23,40)(15,27,28,34)(16,30,31,32)(19,37,24,42)(20,21,38,29)(25,43,26,44)(35,39,41,36)Polycyclic presentation:
2^9|a^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913670);