Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
988 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)(6,10)(7,12)(8,11)(13,14,17,21,18,15,19,23)(16,20,25,27,24,22,26,28);(5,6,11,12,9,7,8,10)(13,15,17,23,18,14,19,21)(16,22,25,28,24,20,26,27);(2,4)(5,7,8,12,9,6,11,10)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(14,15)(16,24)(21,23)(25,26)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ch,d^b=di,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72343950, #72343954, #72343977 |
| maximal normal subgroup of | #72343950, #72343954, #72343977 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109886);