Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,11,9,12,7,10)(14,19,21,24)(15,22,23,25);(1,2,3,4)(6,10)(7,8)(11,12)(13,14,16,19,18,21,17,24)(15,20,22,26,23,28,25,27);(6,11)(7,8)(10,12)(13,15)(14,20)(16,22)(17,25)(18,23)(19,26)(21,28)(24,27);(5,7,9,8)(6,10,12,11)(13,16,18,17)(14,19,21,24)(15,22,23,25)(20,26,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^b=di,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,f^c=fh,g^c=gh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72343943, #72343945, #72343951, … |
| maximal normal subgroup of | #72343943, #72343945, #72343951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100886);