Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,7,10,15,11,16)(8,19,22,17,23,13,21,30)(9,24,28,25,26,14,27,18)(20,32,35,31,33,29,34,36);(2,4)(5,7,12,15,10,16,11,6)(8,19,22,17,23,13,21,30)(9,25,28,14,26,18,27,24)(20,29,35,36,33,32,34,31);(1,2,3,4)(5,8,12,21,10,23,11,22)(6,13,7,17,15,19,16,30)(9,20,28,34,26,33,27,35)(14,29,18,31,24,32,25,36);(5,9)(6,14)(7,18)(8,20)(10,26)(11,27)(12,28)(13,29)(15,24)(16,25)(17,31)(19,32)(21,34)(22,35)(23,33)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^b=fi,e^c=ei,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308196 |
| maximal normal subgroup of | #72308196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275441);