Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130490 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #43001 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×D42 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
1209 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 1209 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10)(11,18)(12,22,23,16)(13,17)(14,19,15,20)(21,24);(2,6)(3,4)(7,8)(9,11)(10,14)(12,19)(13,21)(15,17)(16,18)(20,23)(22,24);(9,11)(10,15)(12,20)(13,21)(14,17)(16,18)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fg,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72230977, #72230987, #72231096, … |
| maximal normal subgroup of | #72230977, #72230987, #72231096, … |
| maximal quotient of | #72240316, #72282655 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37952);