Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2228 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)(7,9)(8,10)(11,12)(13,14)(15,20)(16,22)(17,21)(18,23)(19,25)(24,28)(26,27);(5,7)(6,9)(8,11)(10,12)(13,15,18,24)(14,19,23,27)(16,25,17,26)(20,22,28,21);(1,2,3,4)(7,11)(9,12)(15,24)(19,27)(20,28)(25,26);(7,11)(9,12)(13,15)(14,20)(16,26)(17,25)(18,24)(19,21)(22,27)(23,28);(6,10)(9,12)(13,16,18,17)(14,21,23,22)(15,25,24,26)(19,20,27,28)Polycyclic presentation:
2^9|b^2=i,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380323);