Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2495 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
714 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)(5,6,12,16,11,17,10,18)(7,19,22,29,21,15,9,13)(8,14,26,32,27,33,25,34)(20,30,23,36,35,31,24,28);(5,7,10,9,11,21,12,22)(6,13,16,19,17,29,18,15)(8,23,26,35,27,24,25,20)(14,30,34,28,33,31,32,36);(1,2,3,4)(5,8)(6,14,17,33)(7,20)(9,24)(10,25)(11,27)(12,26)(13,28,29,36)(15,31,19,30)(16,32,18,34)(21,35)(22,23);(5,9,10,21,11,22,12,7)(6,15,16,13,17,19,18,29)(8,24,25,35,27,23,26,20)(14,31,32,28,33,30,34,36)Polycyclic presentation:
2^9|a^2=h,b^2=egh,c^2=efg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277663);