Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455699 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2797 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16893 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
907 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)(3,12,16,24)(4,17)(5,10)(6,21)(7,11)(8,15,13,22)(9,20)(14,28,30,23)(18,25)(19,26)(27,31,32,29);(1,3)(2,8,11,13)(4,14)(5,15)(6,22)(7,16)(9,23,25,28)(10,12,21,24)(17,32,26,27)(18,29)(19,30)(20,31);(1,2)(3,13)(4,17,19,26)(5,10)(6,21)(7,11)(8,16)(9,18,25,20)(12,22)(14,27,30,32)(15,24)(23,31,28,29);(1,4)(2,9)(3,14)(5,20)(6,18)(7,19)(8,23)(10,26)(11,25)(12,27)(13,28)(15,31)(16,30)(17,21)(22,29)(24,32)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318824, #72318890, #72318910, … |
| maximal normal subgroup of | #72318824, #72318890, #72318910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381213);