Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
3266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,18)(16,17)(19,20)(22,25)(26,28);(2,4)(6,9)(10,11)(13,14,16,17)(15,19,20,18)(21,22,24,25)(23,26,27,28);(1,2,3,4)(7,12)(10,11)(14,17)(18,19)(23,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,17)(18,20)(21,23,24,27)(22,26,25,28);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,28)(24,25)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gh,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72332221, #72332234, #72332270 |
| maximal normal subgroup of | #72332221, #72332234, #72332270 |
| maximal quotient of | #72333075 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3068419);