Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(5,6,8,7)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18);(12,13)(16,21)(18,23)(19,22);(5,7,8,6)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,f^b=fi,d^c=di,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72257937, #72264524, #72264560, … |
| maximal normal subgroup of | #72257937, #72264524, #72264560, … |
| maximal quotient of | #72265491, #72279520, #72308415 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262021);