Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3562 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3332 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)(3,4)(13,14,19,23,18,22,17,24)(15,25,34,20,27,36,28,37)(16,29,32,21,31,40,33,39)(26,35,42,41,30,38,43,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,30)(17,34)(18,27)(19,28)(21,38)(22,37)(23,25)(24,36)(26,31)(29,44)(32,42)(33,43)(35,39)(40,41);(6,9)(7,11)(13,16)(14,21)(15,26)(17,32)(18,31)(19,33)(20,35)(22,39)(23,29)(24,40)(25,41)(27,30)(28,42)(34,43)(36,44)(37,38);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(16,31)(23,24)(28,34)(32,33)(35,38)(41,44);(6,9)(10,12)(13,17,18,19)(14,23,22,24)(15,28,27,34)(16,32,31,33)(20,36,37,25)(21,29,39,40)(26,42,30,43)(35,44,38,41)Polycyclic presentation:
2^9|a^2=gi,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6424032);