Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
3326 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(22,24)(23,27);(6,7)(10,11)(13,14,17,16)(15,18,20,19)(23,27)(26,28);(8,12)(10,11)(14,16)(18,19)(21,22,25,24)(23,26,27,28);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,17,20)(14,18,16,19)(21,23,25,27)(22,26,24,28);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,18,20,19)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72332198, #72332221, #72332261 |
| maximal normal subgroup of | #72332198, #72332221, #72332261 |
| maximal quotient of | #72332982 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3271553);