Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 | #2563 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1000 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)(7,17,21,26)(8,22)(9,12)(10,15)(11,16)(13,20,28,18)(14,23)(19,33,35,29)(24,30)(25,27)(31,34,36,32);(6,12,16,15)(7,18,21,20)(13,28)(14,27,30,22)(17,26)(19,32,35,34)(29,33)(31,36);(1,2,3,4)(5,7,10,20,11,21,9,18)(6,13,12,26,16,28,15,17)(8,19,24,34,25,35,23,32)(14,29,27,36,30,33,22,31);(5,8,9,23,11,25,10,24)(6,14,15,22,16,30,12,27)(7,19,18,32,21,35,20,34)(13,29,17,31,28,33,26,36)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=efgh,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282762);