Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479323 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17199 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
660 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)(3,13)(4,18,21,10)(5,11,6,9)(8,14)(15,28)(16,24)(17,23)(19,25,20,27)(22,29)(26,30)(31,32);(1,3)(2,8,12,24)(4,15)(5,16)(6,14)(7,17)(9,23,11,13)(10,26,18,22)(19,30)(20,29)(21,31)(25,28,27,32);(2,9)(3,14)(4,19)(5,6)(11,12)(13,23)(16,17)(20,21)(22,28)(25,27)(26,32)(29,30);(1,4,6,20,7,21,5,19)(2,10,11,27,12,18,9,25)(3,15,14,29,17,31,16,30)(8,22,13,28,24,26,23,32)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386519);