Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4164 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)(6,7)(8,12)(13,14,18,16)(15,17,20,19)(22,25)(23,27);(5,6,9,7)(8,10,12,11)(13,15,18,20)(14,19,16,17)(21,22,24,25)(23,26,27,28);(8,12)(10,11)(15,20)(17,19)(21,23,24,27)(22,26,25,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,17,20,19)(21,22)(23,26)(24,25)(27,28);(5,8)(6,10)(7,11)(9,12)(13,17)(14,15)(16,20)(18,19)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fg,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 | #72323440, #72323441, #72323450, … |
| maximal normal subgroup of | #72323440, #72323441, #72323450, … |
| maximal quotient of | #72323659, #72332517 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 896619);