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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2311 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)(15,26)(16,31)(17,23)(18,24)(19,25)(20,35)(21,29)(22,32)(27,40)(28,42)(30,38)(33,37)(34,39)(36,44)(41,43);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,27)(17,33)(18,28)(19,29)(20,30)(22,36)(23,40)(24,38)(25,26)(31,37)(32,43)(34,44)(35,42)(39,41);(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,27)(17,19)(18,32)(20,34)(21,36)(23,24)(25,39)(26,41)(28,43)(29,33)(30,44)(31,35)(37,42)(38,40);(1,3,4,2)(7,12)(9,11)(13,17,19,16)(14,23,25,31)(15,27,29,33)(18,32,20,34)(21,37,26,40)(22,35,39,24)(28,44,30,43)(36,38,41,42);(6,10)(9,11)(13,18,19,20)(14,24,25,35)(15,28,29,30)(16,32,17,34)(21,38,26,42)(22,23,39,31)(27,43,33,44)(36,40,41,37)Polycyclic presentation:
2^9|a^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6903724);