Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16747 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
664 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)(4,16,18,31)(5,20)(6,10)(7,11)(8,15)(9,19,25,17)(13,24)(14,23,27,32)(21,22)(26,29,30,28);(1,3,6,15,7,13,5,21)(2,8,10,24,11,22,20,12)(4,14,19,29,18,27,17,28)(9,23,16,30,25,32,31,26);(3,13)(4,17)(5,6)(8,22)(9,16)(10,20)(14,28)(18,19)(23,30)(25,31)(26,32)(27,29);(1,4,5,17,7,18,6,19)(2,9,10,16,11,25,20,31)(3,14,21,28,13,27,15,29)(8,23,24,30,22,32,12,26)Polycyclic presentation:
2^9|a^2=fg,b^2=h,d^2=ef,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,d^c=de,e^c=ei,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251684, #72251695, #72251703, … |
| maximal normal subgroup of | #72251684, #72251695, #72251703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378277);