Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2358 |
| prime radical | #18516 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,12,17,13,18,22,16)(11,15,20,23,21,24,19,14);(9,11,12,20,13,21,22,19)(10,14,17,15,18,23,16,24);(2,6)(3,4)(7,8);(1,3,5,4)(2,7,8,6)(9,11)(10,15)(12,20)(13,21)(14,16)(17,23)(18,24)(19,22)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=de,e^b=eh,d^c=di,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72309531, #72342635, #72342639 |
| maximal normal subgroup of | #72309531, #72342635, #72342639 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 135359);