Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3547 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,22)(16,29)(17,25)(18,24)(19,23)(20,28)(21,30)(26,40)(27,35)(31,39)(32,38)(33,37)(34,41)(36,43)(42,44);(5,7)(6,9)(8,12)(10,11)(13,15,18,28)(14,20,24,22)(16,26,31,42)(17,32,19,27)(21,34,37,36)(23,38,25,35)(29,44,39,40)(30,43,33,41);(2,4)(13,16,19,30,18,31,17,33)(14,21,25,29,24,37,23,39)(15,26,27,43,28,42,32,41)(20,34,35,44,22,36,38,40);(7,12)(9,11)(13,15)(14,22)(16,26)(17,32)(18,28)(19,27)(20,24)(21,36)(23,35)(25,38)(29,40)(30,43)(31,42)(33,41)(34,37)(39,44);(6,10)(9,11)(13,17,18,19)(14,23,24,25)(15,27,28,32)(16,30,31,33)(20,35,22,38)(21,29,37,39)(26,41,42,43)(34,40,36,44)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=ei,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300707);