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 | |
2328 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)(15,20)(16,26)(17,22)(18,32)(19,23)(21,24)(25,35)(27,42)(28,34)(29,43)(30,39)(31,44)(33,37)(36,40)(38,41);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,23,32)(16,24,28,33)(17,25,30,31)(21,34,37,26)(22,35,39,44)(27,40,29,41)(36,43,38,42);(5,7)(6,9)(8,12)(10,11)(13,16,19,28)(14,21,23,37)(15,24,18,33)(17,29,30,27)(20,34,32,26)(22,38,39,36)(25,41,31,40)(35,42,44,43);(1,3,4,2)(7,12)(9,11)(13,17,19,30)(14,22,23,39)(15,25,18,31)(16,27,28,29)(20,35,32,44)(21,36,37,38)(24,40,33,41)(26,42,34,43);(6,10)(9,11)(17,30)(22,39)(25,31)(27,29)(35,44)(36,38)(40,41)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,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, 6903721);