Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,16,17)(15,19,20,18)(21,22,25,28)(23,24,27,26);(2,4)(6,9)(10,12)(14,17)(15,20)(21,23,25,27)(22,26,28,24);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,16,20)(14,18,17,19)(21,24,25,26)(22,27,28,23);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,23)(25,26)(27,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=f,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 | #72232319, #72232348, #72232399, … |
| maximal normal subgroup of | #72232319, #72232348, #72232399, … |
| maximal quotient of | #72323676, #72329940, #72330207 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1448875);