Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14373 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,22)(3,12,15,8,16,23,13,25)(4,17,19,9,21,27,20,28)(14,30,32,29,31,24,18,26);(1,3)(2,8)(4,18)(5,13)(6,15)(7,16)(9,26)(10,23)(11,25)(12,22)(14,19)(17,24)(20,31)(21,32)(27,30)(28,29);(3,13,16,15)(8,23,25,12)(14,18,31,32)(24,26,30,29);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,24)(10,27)(11,28)(12,29)(13,18)(15,32)(16,31)(17,22)(23,26)(25,30)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338618, #72338625, #72338635 |
| maximal normal subgroup of | #72338618, #72338625, #72338635 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291369);