Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186967 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,13,16)(3,8,11,15,4,7,12,14)(18,22,24,27)(23,31,32,30);(1,3,6,4)(2,7,9,8)(5,12,13,11)(10,15,16,14)(17,18)(19,23)(20,27)(21,24)(22,29)(25,30)(26,32)(28,31);(1,4,6,3)(2,8,9,7)(5,11,13,12)(10,14,16,15)(17,19,21,26)(18,23,24,32)(20,28,29,25)(22,30,27,31);(2,9)(7,8)(10,16)(14,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268184, #72268187, #72268191, … |
| maximal normal subgroup of | #72268184, #72268187, #72268191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94246);