Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14401 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1143 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)(3,14,19,32)(4,20)(5,23,25,10)(6,11)(7,13)(8,17,28,18)(9,21)(12,26)(15,31,24,27)(16,30)(22,29);(1,3)(2,8)(4,5)(6,18)(7,19)(9,10)(11,32)(12,14)(13,28)(15,16)(17,26)(20,27)(21,24)(22,25)(23,30)(29,31);(1,4,6,21,7,22,26,16)(2,9,12,20,13,30,11,29)(3,15,18,5,19,24,17,25)(8,27,14,23,28,31,32,10);(1,5,6,24,7,25,26,15)(2,10,12,27,13,23,11,31)(3,16,18,4,19,21,17,22)(8,20,14,30,28,29,32,9)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213889, #72219488, #72219493, … |
| maximal normal subgroup of | #72213889, #72219488, #72219493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292350);