Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2778 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15376 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,5,7,24)(3,12,14,23,16,26,15,9)(4,17,20,19,22,28,21,10)(13,30,31,29,32,25,18,27);(1,3)(2,9)(4,18)(5,23)(6,15)(7,14)(8,16)(10,27)(11,26)(12,24)(13,21)(17,25)(19,29)(20,32)(22,31)(28,30);(1,4)(2,10)(3,13)(5,19)(6,21)(7,20)(8,22)(9,25)(11,28)(12,29)(14,18)(15,31)(16,32)(17,24)(23,30)(26,27);(1,5,8,2)(3,12)(4,19,22,10)(6,24,7,11)(9,15)(13,29)(14,23)(16,26)(17,20,28,21)(18,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285398, #72285925, #72286050, … |
| maximal normal subgroup of | #72285398, #72285925, #72286050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335188);