Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10,14,18)(11,15,13,17)(12,16,21,24)(19,23,20,22);(1,2)(3,4)(5,7,8,6)(9,11,14,13)(10,15,18,17)(12,19,21,20)(16,22,24,23);(2,4)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24);(9,13,14,11)(10,17,18,15)(12,20,21,19)(16,23,24,22)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280663, #72302785, #72302791, … |
| maximal normal subgroup of | #72280663, #72302785, #72302791, … |
| maximal quotient of | #72302172, #72302173 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271710);