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 | |
3309 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)(7,9)(8,11)(10,12)(14,17)(18,19)(21,22,25,27)(23,28,26,24);(1,2,3,4)(6,7)(8,12)(13,14)(15,19)(16,17)(18,20)(21,23,25,26)(22,24,27,28);(1,2)(3,4)(6,7)(10,11)(14,17)(18,19)(21,24,25,28)(22,26,27,23);(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,28)(26,27);(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=fgh,c^2=fi,d^2=fgi,e^2=fgi,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 | #72327186, #72327223, #72327280, … |
| maximal normal subgroup of | #72327186, #72327223, #72327280, … |
| maximal quotient of | #72332778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1773748);