Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #374120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14134 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12)(4,18,21,26)(5,7,14,10)(8,16)(9,20,25,19)(13,22)(15,23)(17,24)(27,29)(28,32)(30,31);(2,7)(3,13)(5,14)(8,22)(9,18)(10,11)(12,24)(15,28)(19,20)(23,31)(25,26)(27,32);(1,3,14,17,6,13,5,16)(2,8,10,12,11,24,7,22)(4,15,19,29,21,28,20,30)(9,23,26,27,25,32,18,31);(1,4)(2,9)(3,15)(5,20)(6,21)(7,18)(8,23)(10,26)(11,25)(12,27)(13,28)(14,19)(16,30)(17,29)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72306313, #72306456, #72306472, … |
| maximal normal subgroup of | #72306313, #72306456, #72306472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281316);