Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
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 | |
3936 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)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,27)(24,25)(26,28);(6,10)(7,12)(13,15,17,20)(14,18,19,16)(21,23,26,24)(22,27,28,25);(7,12)(8,11)(14,19)(16,18)(22,28)(23,24);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,23)(22,27,28,25);(5,8,9,11)(6,7,10,12)(13,16,17,18)(14,15,19,20)(21,25,26,27)(22,23,28,24)Polycyclic presentation:
2^9|b^2=g,d^2=fi,e^2=fh,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 | #72300371, #72300384, #72300436, … |
| maximal normal subgroup of | #72300371, #72300384, #72300436, … |
| maximal quotient of | #72330523, #72331574 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4989958);