Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #463891 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16495 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
851 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)(3,14)(4,9)(5,20,23,10)(6,11,18,13)(8,17)(15,19)(16,24)(21,30)(22,28,31,27)(25,29)(26,32);(1,3,6,16,7,9,18,17)(2,8,13,4,12,24,11,14)(5,15,22,30,23,25,31,26)(10,21,28,19,20,32,27,29);(1,4,18,8,7,14,6,24)(2,9,11,16,12,3,13,17)(5,21,31,29,23,32,22,19)(10,26,27,25,20,30,28,15);(1,5)(2,10,12,20)(3,15)(4,19,14,29)(6,22)(7,23)(8,21,24,32)(9,25)(11,27,13,28)(16,30)(17,26)(18,31)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=efi,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251690, #72251691, #72251706, … |
| maximal normal subgroup of | #72251690, #72251691, #72251706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371088);