Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(2,4)(13,15)(14,19)(16,26)(17,24)(18,25)(20,28)(21,23)(22,27);(6,9)(7,12)(10,11)(15,24)(16,17)(19,23)(20,21)(25,26)(27,28);(15,25)(19,27)(23,28)(24,26)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=eh,f^c=fi,g^c=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72310302, #72310309, #72310325, … |
| maximal normal subgroup of | #72310302, #72310309, #72310325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140252);