Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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×#3546 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,20,23,19,16,18,24)(15,25,17,22,26,21,27,28);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,21,16,25)(17,18,27,20)(22,23,28,24);(1,2,3,4)(13,16,19,14)(15,25,26,21)(17,28,27,22)(18,24,20,23);(7,12)(9,11)(13,17)(14,22)(15,18)(16,28)(19,27)(20,26)(21,23)(24,25);(6,10)(9,11)(13,18,19,20)(14,23,16,24)(15,17,26,27)(21,22,25,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ghi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,g^c=gi,h^c=hi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300645);