Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150560 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11796 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| prime radical | #23444 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
710 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,15,17,8)(5,18)(6,10)(7,13,21,28)(9,19)(12,25,27,24)(14,23)(16,22,31,30)(20,26)(29,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,23)(10,21)(11,18)(15,29)(16,26)(17,27)(19,28)(22,25)(24,30)(31,32);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,24)(13,26)(14,27)(18,30)(19,31)(21,29)(23,25)(28,32)Polycyclic presentation:
2^9|a^2=e,d^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efg,h^b=hi,e^c=ei,e^d=eh,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72237594, #72237721, #72237722, … |
| maximal normal subgroup of | #72237594, #72237721, #72237722, … |
| maximal quotient of | #72338681, #72341199 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57756);