Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455816 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16270 |
| 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 | |
1128 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)(4,19)(5,11)(6,10)(7,12)(8,16)(9,20)(14,30)(15,23)(17,25)(18,26)(21,27)(22,28)(24,31)(29,32);(1,3,5,17,7,16,6,18)(2,8,10,25,12,13,11,26)(4,15,20,14,22,32,21,31)(9,24,19,29,27,30,28,23);(1,4)(2,9)(3,14,16,31)(5,21)(6,20)(7,22)(8,23,13,29)(10,28)(11,19)(12,27)(15,18,32,17)(24,25,30,26);(1,4)(2,9)(3,15)(5,20)(6,21)(7,22)(8,24)(10,19)(11,28)(12,27)(13,30)(14,17)(16,32)(18,31)(23,26)(25,29)Polycyclic presentation:
2^9|b^2=f,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,f^c=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251786, #72251791, #72251800, … |
| maximal normal subgroup of | #72251786, #72251791, #72251800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363012);