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) = C26×C4 |
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:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,19)(16,17)(18,20)(22,25)(26,28);(5,7,9,8)(6,10,11,12)(13,15,18,16)(14,19,20,17)(21,22,24,25)(23,26,27,28);(7,8)(10,12)(14,20)(15,16)(23,27)(26,28);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,16,18,15)(14,17,20,19)(21,23,24,27)(22,26,25,28);(5,6)(7,12)(8,10)(9,11)(13,17)(14,15)(16,20)(18,19)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=fg,d^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 | #72327232, #72327245, #72327296, … |
| maximal normal subgroup of | #72327232, #72327245, #72327296, … |
| maximal quotient of | #72330792 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4848005);