Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394483 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2472 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14817 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1065 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)(3,8)(4,16,18,29)(5,21)(6,11)(7,13)(9,19,28,20)(10,23)(12,24)(14,17,27,32)(15,22)(25,30,26,31);(1,3)(2,8)(4,14,18,27)(5,12)(6,15)(7,10)(9,26,28,25)(11,22)(13,23)(16,32,29,17)(19,30,20,31)(21,24);(1,4,6,20,7,18,24,19)(2,9,11,29,13,28,12,16)(3,14,15,31,10,27,21,30)(5,17,8,25,22,32,23,26);(1,5,6,8,7,22,24,23)(2,10,11,21,13,3,12,15)(4,17,20,25,18,32,19,26)(9,27,29,30,28,14,16,31)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284248 |
| maximal normal subgroup of | #72284248 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308672);