Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469776 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1522 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)(7,15,18,19)(8,11,17,12)(9,13)(10,14)(16,20);(1,2,3,4)(5,7,16,8,10,18,9,17)(6,11,13,15,14,12,20,19);(2,4)(5,6)(7,12)(8,19)(9,20)(10,14)(11,18)(13,16)(15,17);(5,8,9,7,10,17,16,18)(6,12,13,19,14,11,20,15)Polycyclic presentation:
2^9|a^2=f,b^2=egh,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286449, #72286979, #72287049, … |
| maximal normal subgroup of | #72286449, #72286979, #72287049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379225);