Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #390446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14572 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,6,8,5,11)(3,13,16,9,17,25,15,22)(4,18,19,10,20,26,21,23)(14,28,29,27,30,24,31,32);(2,8)(5,7)(13,25)(15,16)(18,26)(19,21)(24,28)(29,31);(1,3,5,16,6,17,7,15)(2,9,11,25,8,22,12,13)(4,14,21,29,20,30,19,31)(10,24,18,32,23,28,26,27);(1,4)(2,10)(3,14)(5,21)(6,20)(7,19)(8,23)(9,24)(11,18)(12,26)(13,27)(15,31)(16,29)(17,30)(22,28)(25,32)Polycyclic presentation:
2^9|a^2=h,c^2=egh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72286681, #72287286, #72287354, … |
| maximal normal subgroup of | #72286681, #72287286, #72287354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297644);