Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,13)(9,14,21,12)(10,15)(16,24)(17,23)(18,22,30,25)(19,27,32,26)(20,28)(29,34)(31,33,36,35);(1,2,3,4)(5,7)(6,11)(8,17)(9,18)(10,16)(12,25)(13,23)(14,22)(15,24)(19,31)(20,29)(21,30)(26,35)(27,33)(28,34)(32,36);(6,12)(7,16)(9,21)(11,22)(13,26)(14,15)(17,29)(19,32)(23,33)(24,25)(27,28)(34,35);(5,8,10,20)(6,13,15,28)(7,17,16,29)(9,19,21,32)(11,23,24,34)(12,26,14,27)(18,31,30,36)(22,33,25,35)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,f^c=fi,g^c=ghi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72293182, #72293185, #72293186, … |
| maximal normal subgroup of | #72293182, #72293185, #72293186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400906);