Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456662 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16284 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,11,7,3)(4,13,17,15,19,27,18,9)(5,10,20,26,21,25,22,14)(16,29,30,23,31,28,32,24);(1,3,6,11,8,12,7,2)(4,15,17,13,19,9,18,27)(5,14,20,25,21,26,22,10)(16,24,30,28,31,23,32,29);(1,4)(2,9)(3,13)(5,16)(6,18)(7,17)(8,19)(10,24)(11,15)(12,27)(14,29)(20,32)(21,31)(22,30)(23,25)(26,28);(1,5)(2,10)(3,14)(4,16)(6,20)(7,22)(8,21)(9,23)(11,25)(12,26)(13,28)(15,24)(17,30)(18,32)(19,31)(27,29)Polycyclic presentation:
2^9|a^2=f,b^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285925, #72286235, #72286246, … |
| maximal normal subgroup of | #72285925, #72286235, #72286246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363858);