Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
3121 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)(7,9)(8,10)(11,12)(13,14,17,16)(15,18,19,20)(21,22)(23,28)(24,25)(26,27);(8,12)(10,11)(14,16)(15,19)(21,23)(22,24)(25,26)(27,28);(6,7)(10,11)(14,16)(18,20)(22,27)(23,25);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,19)(14,18,16,20)(21,24,26,28)(22,23,27,25);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,20,19,18)(21,25,26,23)(22,28,27,24)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72300476, #72300513, #72300553 |
| maximal normal subgroup of | #72300476, #72300513, #72300553 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4949083);