Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478349 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17142 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,7,16,21)(3,8,11,24,13,19,17,22)(4,9,14,26,15,20,18,27)(12,25,29,32,30,23,28,31);(2,7)(3,11)(5,16)(8,22)(9,20)(12,29)(13,17)(14,18)(19,24)(23,32)(25,31)(28,30);(1,3)(2,8,7,19)(4,12)(5,17)(6,13)(9,25,20,23)(10,22,21,24)(11,16)(14,28)(15,30)(18,29)(26,31,27,32);(1,4)(2,9,7,20)(3,12)(5,18)(6,15)(8,23,19,25)(10,27,21,26)(11,28)(13,30)(14,16)(17,29)(22,31,24,32)Polycyclic presentation:
2^9|a^2=g,c^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72338617, #72338620, #72338632 |
| maximal normal subgroup of | #72338617, #72338620, #72338632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385545);