Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
4149 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)(7,11)(8,9)(10,12)(14,15)(16,20)(21,22)(23,26)(24,25)(27,28);(5,6,9,8)(7,10,12,11)(14,15)(18,19)(22,25)(26,27);(6,8)(10,11)(13,14,17,15)(16,18,20,19)(21,22)(23,27)(24,25)(26,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,14)(16,19,20,18)(21,22)(23,27)(24,25)(26,28);(5,8,9,6)(7,10,12,11)(13,16,17,20)(14,18,15,19)(21,23,24,28)(22,26,25,27)Polycyclic presentation:
2^9|b^2=fgh,c^2=fh,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72300371, #72300372, #72300455, … |
| maximal normal subgroup of | #72300371, #72300372, #72300455, … |
| maximal quotient of | #72330858, #72331306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3270711);