Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114647 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,26,29,23)(20,22)(21,25)(24,30)(27,31,28,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,22)(19,27)(20,30)(24,25)(28,29)(31,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,29)(18,23,25,26)(20,28,30,27)(22,31,24,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,f^b=fi,g^b=ghi,f^c=fi,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312362, #72312374, #72312383, … |
| maximal normal subgroup of | #72312362, #72312374, #72312383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 21844);