Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,19,23)(15,24,22,18)(16,20,26,28)(17,21,27,25);(5,6)(7,11)(8,10)(9,12)(13,15,18,14,19,22,24,23)(16,25,27,20,26,21,17,28);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,20)(15,25)(17,24)(18,27)(19,26)(21,22)(23,28);(7,12)(9,11)(13,17,19,27)(14,21,23,25)(15,20,22,28)(16,24,26,18);(6,10)(9,11)(16,26)(17,27)(20,28)(21,25)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407667);