Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,25)(15,21,28,39)(16,29,32,22)(17,23,20,26)(18,33,35,24)(27,38,36,43)(30,42,34,40)(31,44,37,41);(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,30)(17,27)(18,31)(19,28)(20,36)(22,40)(23,38)(24,41)(25,39)(26,43)(29,42)(32,34)(33,44)(35,37);(1,2,3,4)(13,16,19,32)(14,22,25,29)(15,18,28,35)(17,31,20,37)(21,24,39,33)(23,41,26,44)(27,34,36,30)(38,42,43,40);(6,10)(9,11)(13,17,19,20)(14,23,25,26)(15,27,28,36)(16,31,32,37)(18,34,35,30)(21,38,39,43)(22,41,29,44)(24,42,33,40);(7,12)(9,11)(13,18,19,35)(14,24,25,33)(15,16,28,32)(17,30,20,34)(21,22,39,29)(23,40,26,42)(27,37,36,31)(38,44,43,41)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590107);