Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13957 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1046 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)(3,11,13,24)(4,16,18,30)(5,9)(6,10)(7,15,20,25)(8,17,21,26)(12,22)(14,28)(19,27)(23,31)(29,32);(1,3,12,25,6,13,5,15)(2,7,9,11,10,20,22,24)(4,14,26,31,18,27,17,29)(8,19,16,23,21,28,30,32);(3,13)(14,27)(15,25)(29,31);(1,4,6,18)(2,8)(3,14,13,27)(5,17,12,26)(7,19)(9,16)(10,21)(11,23)(15,29,25,31)(20,28)(22,30)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72216077, #72216238, … |
| maximal normal subgroup of | #72213409, #72216077, #72216238, … |
| maximal quotient of | #72210978, C2×#363996, #72239210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271192);