Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198171 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#3205 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8,6,10,4,9)(3,11,13,16,14,15,12,7)(17,18)(19,22,26,31)(20,27)(21,24)(23,29)(25,32,28,30);(1,3)(2,7,10,16)(4,12)(5,13)(6,14)(8,11,9,15)(17,19,21,26)(18,22)(20,28,29,25)(23,32)(24,31)(27,30);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)(20,29)(23,27)(25,28)(30,32);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,15,16,11)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,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 | #72306532, #72306542, #72306555, … |
| maximal normal subgroup of | #72306532, #72306542, #72306555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105676);