Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3182 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)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,17)(18,20)(22,23)(26,27);(7,11)(10,12)(14,16)(15,20)(21,22)(23,25)(24,26)(27,28);(6,9)(10,12)(14,16)(18,19)(22,23)(24,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,16,19)(21,23,25,22)(24,27,28,26);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|a^2=fgh,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72334020, #72334026, #72334043 |
| maximal normal subgroup of | #72334020, #72334026, #72334043 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5192324);