Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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×#3100 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,7)(8,9)(11,12)(13,14,16,18,17,21,23,20)(15,22,25,19,24,26,28,27);(1,2,3,4)(5,6,9,11,10,12,8,7)(14,18,21,20)(19,26,27,22);(5,7,8,12,10,11,9,6)(13,15,23,28,17,24,16,25)(14,19,18,26,21,27,20,22);(15,24)(19,27)(22,26)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72215888, #72216042, #72216045, … |
| maximal normal subgroup of | #72215888, #72216042, #72216045, … |
| maximal quotient of | #72321784 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111569);