Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3280 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)(10,11)(14,17)(15,20)(21,22)(23,24)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(13,15)(14,18)(16,20)(17,19)(21,22,25,28)(23,24,27,26);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15)(14,19)(16,20)(17,18)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,27)(23,28)(25,26)Polycyclic presentation:
2^9|c^2=fghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232870, #72232901, #72232935, … |
| maximal normal subgroup of | #72232870, #72232901, #72232935, … |
| maximal quotient of | #72328444, #72328505, #72329219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1198625);