Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2537 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16)(7,17,20,26)(8,21,24,27)(9,12,10,15)(13,25,28,18)(14,23,30,22)(19,33,34,29)(31,35,36,32);(6,12,16,15)(7,18,20,25)(13,28)(14,27,30,21)(17,26)(19,32,34,35)(29,33)(31,36);(5,7)(6,13)(8,19)(9,18)(10,25)(11,20)(12,26)(14,29)(15,17)(16,28)(21,31)(22,32)(23,35)(24,34)(27,36)(30,33);(1,3,4,2)(5,8,10,23,11,24,9,22)(6,14,12,27,16,30,15,21)(7,19,25,35,20,34,18,32)(13,29,26,36,28,33,17,31)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286008);