Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
3596 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(22,24)(23,27);(7,11)(10,12)(13,14)(15,19)(16,17)(18,20)(21,22,25,24)(23,26,27,28);(6,9)(10,12)(14,17)(18,19)(23,27)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,22)(23,26,27,28)Polycyclic presentation:
2^9|b^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234676, #72234680, #72297800, … |
| maximal normal subgroup of | #72234676, #72234680, #72297800, … |
| maximal quotient of | #72298153, #72328890, #72328896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5529845);