Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 4194304 |
| 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 | C22×#392 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5294 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)(5,6,9,11)(7,8,12,10)(13,14,16,18)(15,19,20,17)(23,27)(25,28);(5,7,9,12)(6,10,11,8)(13,14,16,18)(15,17,20,19)(21,22,24,26)(23,25,27,28);(6,11)(7,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,20);(5,8)(6,7)(9,10)(11,12)(13,15)(14,17)(16,20)(18,19)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72326273, #72326276, #72326282, … |
| maximal normal subgroup of | #72326273, #72326276, #72326282, … |
| maximal quotient of | #72328382, #72328394, #72330515 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4360454);