Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3573 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)(6,9)(7,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(1,2,3,4)(5,6,8,9)(7,12,11,10)(13,14,16,17)(15,18,20,19)(21,22,24,26)(23,25,28,27);(2,4)(7,11)(10,12)(14,17)(18,19)(21,22,24,26)(23,25,28,27);(2,4)(5,6)(7,12)(8,9)(10,11)(13,15)(14,18)(16,20)(17,19)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,27)(24,26)(25,28)Polycyclic presentation:
2^9|b^2=f,c^2=fhi,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232901, #72232920, #72232968, … |
| maximal normal subgroup of | #72232901, #72232920, #72232968, … |
| maximal quotient of | #72328748, #72328875, #72330588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3908695);