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 | |
3697 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)(5,6,8,9)(7,11,12,10)(13,14,17,19)(15,16,20,18)(21,22)(23,28)(24,26)(25,27);(6,9)(10,11)(13,15,17,20)(14,18,19,16)(22,24)(25,28);(7,12)(10,11)(14,19)(15,20)(21,23,26,27)(22,25,24,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,20)(18,19)(21,24,26,22)(23,25,27,28);(5,6)(7,11)(8,9)(10,12)(13,16)(14,20)(15,19)(17,18)(21,25)(22,27)(23,24)(26,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^2=gh,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 | #72278072, #72300497, #72300503, … |
| maximal normal subgroup of | #72278072, #72300497, #72300503, … |
| maximal quotient of | #72278503, #72299076, #72299624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2984362);