Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14722 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1719 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)(3,13)(4,9)(5,20)(6,22)(7,12)(8,15)(10,21)(11,23)(14,29)(16,28)(17,27)(18,32)(19,26)(24,30)(25,31);(1,3,6,15,7,16,23,31)(2,8,11,13,12,25,22,28)(4,17,18,21,19,14,30,5)(9,10,24,27,26,20,32,29);(1,4)(2,9)(3,14)(5,15)(6,18)(7,19)(8,10)(11,24)(12,26)(13,27)(16,17)(20,25)(21,31)(22,32)(23,30)(28,29);(1,5,6,17,7,21,23,14)(2,10,11,27,12,20,22,29)(3,4,15,18,16,19,31,30)(8,24,13,26,25,32,28,9)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319969, #72320009, #72320106, … |
| maximal normal subgroup of | #72319969, #72320009, #72320106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302386);