Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10,14,19,15,20,13,17)(11,21,18,16,23,24,22,12);(1,2,4,3)(5,7,8,6)(9,11,13,18,15,23,14,22)(10,16,17,24,20,12,19,21);(10,17)(11,22)(13,14)(18,23)(19,20)(21,24);(1,3,4,2)(9,12)(10,18)(11,19)(13,21)(14,24)(15,16)(17,23)(20,22)Polycyclic presentation:
2^9|a^2=h,b^2=egh,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226917, #72226925, #72226944 |
| maximal normal subgroup of | #72226917, #72226925, #72226944 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275097);