Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10,13,17,14,18,12,15)(11,19,21,16,22,23,20,24);(2,6)(3,4)(7,8)(10,15,18,17)(16,19,24,23);(2,6)(3,4)(7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,23)(18,24);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72210583, #72213659, #72213681, … |
| maximal normal subgroup of | #72210583, #72213659, #72213681, … |
| maximal quotient of | #72240395, C2×#198728, #72279080 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105924);