Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473852 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16884 |
| 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 | |
900 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)(3,12)(4,17,19,25)(5,21)(6,9)(7,11)(8,14)(10,20,24,18)(13,23,29,31)(15,22)(16,26)(27,30,32,28);(1,3,5,15,6,16,7,14)(2,8,21,12,9,22,11,26)(4,13,20,28,19,29,18,30)(10,23,17,27,24,31,25,32);(2,9)(4,18)(5,7)(8,22)(10,25)(13,28)(14,15)(17,24)(19,20)(23,27)(29,30)(31,32);(1,4)(2,10)(3,13)(5,20)(6,19)(7,18)(8,23)(9,24)(11,25)(12,27)(14,30)(15,28)(16,29)(17,21)(22,31)(26,32)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251688, #72251695, #72251700, … |
| maximal normal subgroup of | #72251688, #72251695, #72251700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381048);