Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #189517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,21,24,13)(9,25)(10,27)(11,16)(12,20)(14,26)(15,19)(18,32,22,31)(23,30,33,35)(28,34,29,36);(1,2)(3,4)(5,7,10,20,11,9,19,26)(6,12,15,17,16,14,27,25)(8,18,23,34,24,22,33,36)(13,28,30,31,21,29,35,32);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,24)(12,28)(14,29)(15,30)(16,21)(17,31)(19,33)(20,34)(25,32)(26,36)(27,35);(5,9,10,26,11,7,19,20)(6,14,15,25,16,12,27,17)(8,22,23,36,24,18,33,34)(13,29,30,32,21,28,35,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240851, #72240855, #72240862, … |
| maximal normal subgroup of | #72240851, #72240855, #72240862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267388);