Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460970 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2759 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16375 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
843 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)(3,12,14,7)(4,16)(5,9)(6,10)(8,18)(11,21)(13,22,17,25)(15,24,29,23)(19,27)(20,26)(28,31,30,32);(1,3)(2,7)(4,17,18,13)(5,15)(6,14)(8,25,16,22)(9,23)(10,12)(11,24)(19,30,20,28)(21,29)(26,32,27,31);(1,4)(2,8,10,16)(3,13)(5,20)(6,18)(7,22,12,25)(9,27,11,26)(14,17)(15,30)(19,21)(23,32,24,31)(28,29);(2,9,10,11)(7,23,12,24)(8,26,16,27)(22,31,25,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285055, #72286050, #72286206, … |
| maximal normal subgroup of | #72285055, #72286050, #72286206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368166);