Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14572 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,25)(3,14,17,9,18,30,26,29)(4,19,21,16,22,10,27,31)(5,20,23,11,24,15,28,32);(1,3)(2,9)(4,20)(5,19)(6,26)(7,17)(8,18)(10,24)(11,21)(12,30)(13,29)(14,25)(15,22)(16,23)(27,32)(28,31);(1,4)(2,10,13,19)(3,15,18,20)(5,9)(6,27)(7,21)(8,22)(11,17,32,26)(12,31,25,16)(14,28)(23,30)(24,29);(1,5)(2,11)(3,16)(4,9)(6,28)(7,23)(8,24)(10,17)(12,15)(13,32)(14,27)(18,31)(19,26)(20,25)(21,30)(22,29)Polycyclic presentation:
2^9|a^2=h,c^2=egh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237074, #72237078, #72237661, … |
| maximal normal subgroup of | #72237074, #72237078, #72237661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297618);