Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,7,8)(9,10,11,14,12,13,15,16)(17,18)(19,24)(20,23)(21,30)(22,26)(25,28)(27,31)(29,32);(1,2)(3,6)(4,8)(5,7)(10,13)(11,15)(18,23)(19,21)(20,28)(24,25)(26,32)(27,29)(30,31);(1,3,5,4)(2,6,7,8)(17,19,22,21)(18,24,26,30)(20,27,29,28)(23,31,32,25)Polycyclic presentation:
2^9|a^2=e,c^2=fgh,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216299, #72216313, #72216346, … |
| maximal normal subgroup of | #72216299, #72216313, #72216346, … |
| maximal quotient of | #72216293, #72216566, #72225354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31416);