Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3582 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)(13,14,17,19,18,21,16,22)(15,23,24,28,26,27,25,20);(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(6,9)(7,11)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,28)(22,23);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,19,21,22)(15,24,26,25)(20,27,28,23);(6,9)(10,12)(14,21)(19,22)(20,28)(23,27)Polycyclic presentation:
2^9|a^2=g,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6424027);