Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1999 |
| prime radical | C25 | D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,15)(8,20)(9,23)(10,13)(11,17)(12,16)(14,21)(18,27)(19,34)(22,35)(24,33)(25,36)(26,31)(28,30)(29,32);(5,7,16,30,9,19,22,33,10,18,31,36,21,32,8,17)(6,11,20,29,14,25,26,27,13,24,35,34,23,28,12,15);(11,24)(15,27)(25,28)(29,34)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250310, #72250316, #72250324, … |
| maximal normal subgroup of | #72250310, #72250316, #72250324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55515);