Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427822 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15375 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,18)(7,21)(8,15)(10,19)(12,26)(13,20)(17,31)(22,29);(1,2)(3,11)(4,15)(5,19)(6,9)(7,13)(8,16)(10,18)(12,27)(14,24)(17,23)(20,21)(22,32)(25,26)(28,29)(30,31);(1,3)(2,7)(4,12)(5,20)(6,14)(8,22)(9,21)(10,24)(11,19)(13,18)(15,29)(16,26)(17,32)(23,27)(25,30)(28,31);(1,4)(2,8,9,15)(3,12)(5,17)(6,16)(7,22,21,29)(10,23,19,30)(11,25,24,27)(13,28)(14,26)(18,31)(20,32)Polycyclic presentation:
2^9|d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285150, #72285868, #72286235, … |
| maximal normal subgroup of | #72285150, #72285868, #72286235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335018);