Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116016 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,10,22)(3,11)(4,15)(8,23,24,21)(9,20,25,27)(12,14)(16,19,31,29)(17,32);(2,8,9,16)(3,12)(7,19,20,23)(10,24,25,31)(11,14)(13,28)(21,22,29,27)(26,30);(1,2)(3,7)(4,16)(5,9)(6,10)(8,17)(11,22)(12,27)(13,29)(14,20)(15,31)(18,25)(19,26)(21,30)(23,28)(24,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,f^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ehi,d^c=dh,f^c=fh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72232999, #72233126, #72233131, … |
| maximal normal subgroup of | #72232999, #72233126, #72233131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23212);