Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461099 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16379 |
| 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 | |
893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,9)(3,13,17,22,14,8,16,24)(4,18,19,10,20,25,21,26)(15,29,31,23,28,32,30,27);(1,3,5,16,7,14,6,17)(2,8,12,22,11,13,9,24)(4,15,19,30,20,28,21,31)(10,23,25,29,26,27,18,32);(2,9)(3,14)(5,6)(8,22)(10,25)(11,12)(13,24)(15,28)(18,26)(19,21)(23,32)(27,29);(1,4)(2,10)(3,15)(5,19)(6,21)(7,20)(8,23)(9,18)(11,26)(12,25)(13,27)(14,28)(16,30)(17,31)(22,29)(24,32)Polycyclic presentation:
2^9|a^2=fg,b^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72318916, #72318935, #72318991, … |
| maximal normal subgroup of | #72318916, #72318935, #72318991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368295);