Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16357 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1174 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)(3,5,17,23)(4,10)(6,12)(7,14)(8,13)(9,16,22,11)(15,31,26,21)(18,29,25,32)(19,27)(20,28)(24,30);(1,3,6,11,7,17,8,16)(2,9,12,23,14,22,13,5)(4,15,19,29,20,26,24,32)(10,25,27,21,28,18,30,31);(1,4)(2,10)(3,15,17,26)(5,21,23,31)(6,19)(7,20)(8,24)(9,18,22,25)(11,29,16,32)(12,27)(13,30)(14,28);(1,5,6,9,7,23,8,22)(2,11,12,17,14,16,13,3)(4,18,19,31,20,25,24,21)(10,26,27,32,28,15,30,29)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318806, #72318935, #72318940, … |
| maximal normal subgroup of | #72318806, #72318935, #72318940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367318);