Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
4383 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)(6,9)(10,12)(13,14,17,18)(15,19,20,16)(22,24)(23,27);(7,11)(10,12)(14,18)(16,19)(22,24)(26,28);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,20)(18,19)(21,22,25,24)(23,28,27,26);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,18)(17,20)(21,23,25,27)(22,26,24,28);(5,7)(6,10)(8,11)(9,12)(13,16,17,19)(14,20,18,15)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=h,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232582, #72232633, #72232693, … |
| maximal normal subgroup of | #72232582, #72232633, #72232693, … |
| maximal quotient of | #72299034, #72299607, #72299658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5537656);