Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C2 | C2×#1781 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,4,9)(3,13,14,30,16,28,21,32)(5,18,17,29,20,10,22,27)(8,23,26,19,25,31,24,15);(1,3)(2,8)(4,14)(5,19)(6,21)(7,16)(9,24)(10,28)(11,26)(12,25)(13,18)(15,20)(17,23)(22,31)(27,32)(29,30);(5,20)(10,18)(13,28)(15,19)(17,22)(23,31)(27,29)(30,32);(1,4,7,6)(2,9,12,11)(3,14,16,21)(5,17,20,22)(8,24,25,26)(10,27,18,29)(13,30,28,32)(15,31,19,23)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233438, #72234206, #72234207, … |
| maximal normal subgroup of | #72233438, #72234206, #72234207, … |
| maximal quotient of | #72281001, C2×#493402, #72286828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400598);