Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #387433 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2372 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14473 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,14)(4,20,22,26)(5,10,21,27)(6,12)(7,11)(8,13)(9,17)(15,29,31,23)(16,32,30,25)(18,24)(19,28);(1,3)(2,9)(4,16)(5,15)(6,18)(7,17)(8,19)(10,25)(11,28)(12,14)(13,24)(20,29)(21,31)(22,30)(23,26)(27,32);(1,4,6,21,8,22,7,5)(2,10,11,26,13,27,12,20)(3,15,17,30,19,31,18,16)(9,23,14,25,24,29,28,32);(1,5,7,22,8,21,6,4)(2,10,11,26,13,27,12,20)(3,16,18,31,19,30,17,15)(9,23,14,25,24,29,28,32)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279298, #72279345, #72279455, … |
| maximal normal subgroup of | #72279298, #72279345, #72279455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294629);