Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | |
2131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,10)(7,11,8,12)(13,14,17,22)(15,24,27,19)(16,29,31,40)(18,21,33,23)(20,34,38,32)(25,37,43,44)(26,39,28,36)(30,35,42,41);(1,2)(3,4)(6,10)(11,12)(13,15)(14,19)(16,30)(17,27)(18,26)(20,37)(21,36)(22,24)(23,39)(25,34)(28,33)(29,41)(31,42)(32,43)(35,40)(38,44);(13,16)(14,20)(15,25)(17,31)(18,34)(19,35)(21,29)(22,38)(23,40)(24,41)(26,42)(27,43)(28,30)(32,33)(36,37)(39,44);(2,4)(5,7,9,8)(6,11,10,12)(14,21)(15,26)(18,33)(20,29)(22,23)(25,42)(27,28)(30,43)(32,34)(36,39)(37,44)(38,40);(5,8,9,7)(6,11,10,12)(14,22)(16,31)(19,24)(21,23)(25,43)(30,42)(32,34)(36,39)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429190);