Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394813 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2436 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14822 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,14,6,13)(3,15,18,10,17,25,4,9)(5,20,23,27,22,28,21,11)(16,26,31,32,30,24,19,29);(1,3,6,4,7,17,8,18)(2,9,12,15,14,10,13,25)(5,16,21,19,22,30,23,31)(11,24,20,29,27,26,28,32);(1,4,6,17,7,18,8,3)(2,10,12,25,14,9,13,15)(5,19,21,30,22,31,23,16)(11,26,20,32,27,24,28,29);(1,5)(2,11)(3,16)(4,19)(6,21)(7,22)(8,23)(9,24)(10,26)(12,20)(13,28)(14,27)(15,29)(17,30)(18,31)(25,32)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280281 |
| maximal normal subgroup of | #72280281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308920);