Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198153 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3213 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6,10,14)(4,7)(5,8)(9,13,11,15)(12,16)(17,18,21,25,22,26,20,23)(19,27,28,24,30,31,29,32);(1,3,5,10)(2,6)(4,11,12,9)(7,15)(8,14)(13,16)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(4,12)(7,16)(9,11)(13,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20,22,21)(18,23,26,25)(19,28,30,29)(24,31,32,27)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72315937, #72315941, #72316022, … |
| maximal normal subgroup of | #72315937, #72315941, #72316022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105652);