Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,17)(18,20)(22,25)(26,27);(14,16)(18,19)(21,22)(23,27)(24,25)(26,28);(6,9)(10,12)(15,20)(18,19)(22,25)(26,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,23,24,28)(22,26,25,27)Polycyclic presentation:
2^9|a^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72230894, #72232380, #72232396, … |
| maximal normal subgroup of | #72230894, #72232380, #72232396, … |
| maximal quotient of | #72237073, #72278117, #72278119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4848116);