Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2207 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 2207 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,7)(4,8)(9,10,14,13)(11,15,19,18)(12,16,21,20)(17,23,24,22);(1,2,5,3)(4,7,8,6)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24);(9,12)(10,16)(11,17)(13,20)(14,21)(15,22)(18,23)(19,24);(1,3,5,2)(4,6,8,7)(9,13,14,10)(11,18,19,15)(12,20,21,16)(17,23,24,22)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh,g^d=gh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72268289, #72268309, #72268320, … |
| maximal normal subgroup of | #72268289, #72268309, #72268320, … |
| maximal quotient of | #72281558 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253737);