Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,7)(8,11,12,10)(14,15)(18,19)(21,22,24,25)(23,27,28,26);(2,4)(6,7)(8,12)(13,14,17,15)(16,19,20,18)(22,25)(23,28);(1,2)(3,4)(6,7)(10,11)(14,15)(18,19)(21,23,24,28)(22,26,25,27);(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,28)(25,27);(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fgi,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232481, #72232494, #72232527, … |
| maximal normal subgroup of | #72232481, #72232494, #72232527, … |
| maximal quotient of | #72298280, #72328859, #72330708 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1492204);