Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14,20,26)(15,21,31,42)(16,32,28,44)(17,36,27,43)(18,24,19,25)(22,35,39,34)(23,33,38,37)(29,40,30,41);(1,2)(3,4)(6,9)(10,11)(13,15,20,31)(14,21,26,42)(16,33,28,37)(17,34,27,35)(18,29,19,30)(22,43,39,36)(23,32,38,44)(24,40,25,41);(13,16,20,28)(14,22,26,39)(15,27,31,17)(18,35,19,34)(21,38,42,23)(24,32,25,44)(29,33,30,37)(36,41,43,40);(2,4)(13,17,20,27)(14,23,26,38)(15,28,31,16)(18,33,19,37)(21,39,42,22)(24,36,25,43)(29,35,30,34)(32,41,44,40);(5,7,8,12)(6,10,9,11)(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,34,28,35)(17,33,27,37)(21,40,42,41)(22,44,39,32)(23,36,38,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,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=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425879);