Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
4253 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)(8,10)(11,12)(13,14,16,17)(15,19,20,18)(21,22,26,25)(23,28,24,27);(1,2,3,4)(5,6,8,10)(7,9,12,11)(14,17)(15,20)(21,23,26,24)(22,27,25,28);(1,2)(3,4)(13,15)(14,18)(16,20)(17,19)(21,23,26,24)(22,28,25,27);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,20)(17,18)(21,24,26,23)(22,28,25,27);(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,25,26,22)(23,27,24,28)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=i,d^2=fg,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232867, #72232914, #72232945, … |
| maximal normal subgroup of | #72232867, #72232914, #72232945, … |
| maximal quotient of | #72323665, #72323716, #72331307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2929096);