Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2679 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,25)(26,28);(1,2,4,3)(6,10)(7,12)(13,15,17,20)(14,16,18,19)(21,22,25,24)(23,27,28,26);(2,3)(6,10)(8,11)(14,18)(16,19)(22,24)(26,27);(5,7)(6,8)(9,12)(10,11)(13,15)(14,19)(16,18)(17,20)(21,23)(22,26)(24,27)(25,28);(1,3,4,2)(5,8)(6,7)(9,11)(10,12)(13,16,17,19)(14,15,18,20)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^9|b^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72329284, #72329339, #72329409, … |
| maximal normal subgroup of | #72329284, #72329339, #72329409, … |
| maximal quotient of | #72330727 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 987681);