Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190116 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2450 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 2450 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,14,20,21)(11,17,19,24,18,15,22,23);(10,14)(12,20)(15,17)(19,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,14,17)(12,19)(13,18)(16,23,21,24)(20,22);(11,18)(15,17)(19,22)(23,24)Polycyclic presentation:
2^9|a^2=g,c^2=e,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219310, #72219325, #72219340, … |
| maximal normal subgroup of | #72219310, #72219325, #72219340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97343);