Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461157 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16381 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
921 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,11,6,12)(3,8,16,24,14,23,15,22)(4,9,20,26,18,27,19,28)(13,21,17,25,29,31,30,32);(1,3)(2,8,11,23)(4,17,18,30)(5,15)(6,16)(7,14)(9,25)(10,22,12,24)(13,19,29,20)(21,26)(27,32)(28,31);(1,4)(2,9,11,27)(3,13,14,29)(5,19)(6,20)(7,18)(8,21)(10,28,12,26)(15,17,16,30)(22,25)(23,31)(24,32);(2,10)(5,6)(8,22)(9,26)(11,12)(13,29)(15,16)(19,20)(21,25)(23,24)(27,28)(31,32)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251685, #72251691, #72251703, … |
| maximal normal subgroup of | #72251685, #72251691, #72251703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368353);