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 | |
2347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,27)(17,22)(18,23)(19,20)(21,25)(26,37)(28,36)(29,40)(30,34)(31,33)(32,38)(35,39)(41,44)(42,43);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,19)(14,20,23,24)(16,28,30,25)(17,31,32,26)(21,34,36,27)(22,37,38,33)(29,39,42,41)(35,40,44,43);(5,7)(6,9)(8,12)(10,11)(13,16,18,30)(14,21,23,36)(15,25,19,28)(17,29,32,42)(20,27,24,34)(22,35,38,44)(26,39,31,41)(33,40,37,43);(1,3,4,2)(7,12)(9,11)(13,17)(14,22)(15,26)(16,29)(18,32)(19,31)(20,33)(21,35)(23,38)(24,37)(25,39)(27,40)(28,41)(30,42)(34,43)(36,44);(6,10)(9,11)(14,23)(17,32)(20,24)(21,36)(26,31)(27,34)(29,42)(39,41)Polycyclic presentation:
2^9|a^2=fi,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=ehi,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, 6903801);