Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396101 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,10,14,18)(8,13)(9,12)(11,15,20,16)(17,19);(1,2)(3,4)(5,7)(6,10,12,18)(8,16)(9,14)(11,19,20,13)(15,17);(2,4)(8,17)(10,18)(13,19)(15,16);(5,7)(6,11,12,20)(8,15)(9,14)(10,19,18,13)(16,17)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219646, #72219662, #72219665 |
| maximal normal subgroup of | #72219646, #72219662, #72219665 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308263);