Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94184 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C2×C4⋊C4 | C42 |
| Frattini, cosocle | C2×#384 | C22 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G | |
| LECS | G ▷ C2×#384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,19,20,24,22,23,21,15)(12,16)(13,17)(14,18);(2,5,7,6)(9,11)(10,15)(12,21)(13,20)(14,22)(16,19)(17,23)(18,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=hi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,f^c=fi,g^c=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72279265 |
| maximal normal subgroup of | #72279265 |
| maximal quotient of | #72279908 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1392);