Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455836 |
| 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 | |
1134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(5,17)(8,22)(9,25)(10,18)(14,30)(15,19)(20,24)(21,27)(23,26)(31,32);(1,2)(3,8,11,24)(4,9)(5,18)(6,10)(7,17)(12,23,28,32)(13,22,29,20)(14,31,30,26)(15,21)(16,27)(19,25);(1,3,5,13,6,11,17,29)(2,8,7,20,10,24,18,22)(4,14,15,28,16,30,19,12)(9,26,25,32,27,31,21,23);(1,4)(2,9)(3,12)(5,19)(6,16)(7,21)(8,23)(10,27)(11,28)(13,30)(14,29)(15,17)(18,25)(20,31)(22,26)(24,32)Polycyclic presentation:
2^9|b^2=f,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237054, #72237066, #72237067, … |
| maximal normal subgroup of | #72237054, #72237066, #72237067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363032);