Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,27)(16,33)(17,22)(18,26)(19,25)(20,24)(21,30)(23,35)(28,39)(29,43)(31,41)(32,40)(34,44)(36,42)(37,38);(1,2,3,4)(13,15,18,30,19,31,20,32)(14,21,24,27,25,40,26,41)(16,34,35,29,17,37,36,28)(22,39,23,38,33,43,42,44);(6,9)(10,12)(13,16,19,17)(14,22,25,33)(15,28,31,29)(18,35,20,36)(21,38,40,44)(23,26,42,24)(27,43,41,39)(30,34,32,37);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,23,25,42)(15,29,31,28)(18,35,20,36)(21,39,40,43)(22,26,33,24)(27,44,41,38)(30,34,32,37)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334790);