Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(18,22)(23,28)(24,29)(30,32);(1,3)(2,6)(4,8)(5,7)(10,14)(11,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(3,8)(5,6)(11,15)(13,16)(18,22)(19,25)(24,29)(26,31);(2,7)(5,6)(18,22)(23,28)(24,29)(30,32);(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)Polycyclic presentation:
2^9|f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72279724, #72280763, #72293029, … |
| maximal normal subgroup of | #72279724, #72280763, #72293029, … |
| maximal quotient of | #72280944, #72304083, #72304096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999246);