Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,7,13,16,11,14,12,15)(17,18)(19,24)(20,25)(21,30,31,26)(22,27)(23,28,32,29);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,19,22,20)(18,23,27,32)(21,28,31,29)(24,30,25,26);(3,11)(7,14)(12,13)(15,16)(17,19,22,20)(18,24,27,25)(21,28,31,29)(23,30,32,26);(1,4,6,5)(2,8,10,9)(3,12,11,13)(7,15,14,16)(17,20,22,19)(18,25,27,24)(21,29,31,28)(23,26,32,30)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=hi,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ch,f^b=fg,f^e=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72285101, #72285155, #72285186, … |
| maximal normal subgroup of | #72285101, #72285155, #72285186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387777);