Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,18,21,24)(12,22,23,15)(13,17)(14,16)(19,20);(1,3,6,2)(4,7,5,8)(9,11,19,22,14,21,13,15)(10,12,20,18,16,23,17,24);(9,12)(10,15)(11,20)(13,24)(14,23)(16,22)(17,21)(18,19);(2,3)(7,8)(10,16)(11,21)(15,22)(17,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214049, #72214667, #72214788, … |
| maximal normal subgroup of | #72214049, #72214667, #72214788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102225);