Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418871 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2697 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15538 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(4,14,16,18)(7,20)(8,23,15,24)(11,25,26,29)(21,31,30,32)(22,28);(1,2)(3,7)(4,15)(5,17)(6,9)(8,16)(10,19)(11,21)(12,20)(13,28)(14,23)(18,24)(22,27)(25,31)(26,30)(29,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,21)(9,20)(10,22)(14,25)(15,30)(16,26)(17,28)(18,29)(19,27)(23,32)(24,31);(1,4)(2,8,9,15)(3,11)(5,18)(6,16)(7,21,20,30)(10,24,17,23)(12,26)(13,29)(14,19)(22,31,28,32)(25,27)Polycyclic presentation:
2^9|a^2=g,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319908, #72319988, #72320059, … |
| maximal normal subgroup of | #72319908, #72319988, #72320059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342976);