Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461121 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16380 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
889 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)(3,13,17,24)(4,18,21,25)(5,11)(6,9)(7,12)(8,14,22,16)(10,20,26,19)(15,23)(27,28)(29,32)(30,31);(1,3)(2,8,12,22)(4,15)(5,14)(6,16)(7,17)(9,24,11,13)(10,23,26,31)(18,27,25,32)(19,29)(20,28)(21,30);(2,9)(3,14)(4,19)(5,6)(8,22)(10,26)(11,12)(16,17)(20,21)(23,27)(28,29)(31,32);(1,4)(2,10)(3,15)(5,20)(6,19)(7,21)(8,23)(9,25)(11,18)(12,26)(13,27)(14,28)(16,29)(17,30)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72318258, #72318916, #72318969, … |
| maximal normal subgroup of | #72318258, #72318916, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368317);