Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393897 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4096 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 4096 conjugacy classes of subgroups — too many to draw. Only its 296 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 — 4 generators on 32 points:
(1,2)(3,13)(4,17)(5,11)(6,10)(7,12)(8,15)(9,19)(14,22)(16,27)(18,26)(20,25)(21,23)(24,30)(28,29)(31,32);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,24)(10,27)(11,13)(12,23)(17,28)(18,31)(19,29)(20,30)(22,26)(25,32);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,22)(10,17)(11,25)(12,26)(13,28)(15,31)(16,30)(21,29)(23,24)(27,32);(5,6)(10,11)(13,27)(15,21)(17,25)(18,19)(28,32)(29,31)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72235139, #72238714, #72238721, … |
| maximal normal subgroup of | #72235139, #72238714, #72238721, … |
| maximal quotient of | #72235174, #72287102, #72287107 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301093);