Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453808 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16124 |
| 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 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,5,11,6,10)(3,8,16,24,14,25,15,23)(4,17,19,28,21,9,20,27)(13,29,18,26,30,22,31,32);(1,3,6,15,5,14,7,16)(2,8,12,24,11,25,10,23)(4,18,20,13,21,31,19,30)(9,26,27,22,17,32,28,29);(1,4)(2,9,11,17)(3,13,14,30)(5,21)(6,19)(7,20)(8,22)(10,27,12,28)(15,18,16,31)(23,32)(24,26)(25,29);(2,10)(6,7)(8,23)(9,27)(11,12)(15,16)(17,28)(18,31)(19,20)(22,32)(24,25)(26,29)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251685, #72251695, #72251706, … |
| maximal normal subgroup of | #72251685, #72251695, #72251706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361004);