Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,20,16)(22,25)(26,27);(2,4)(6,9)(10,12)(14,18)(16,19)(21,22)(23,27)(24,25)(26,28);(1,2,3,4)(7,11)(10,12)(14,18)(15,20)(22,25)(26,27);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,16,17,19)(14,15,18,20)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=g,c^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232459, #72232502, #72232530, … |
| maximal normal subgroup of | #72232459, #72232502, #72232530, … |
| maximal quotient of | #72329928 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 612611);