Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456381 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2712 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16798 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,11,25,10)(3,12,14,22,15,24,16,5)(4,8,19,27,20,29,23,28)(13,18,30,21,31,26,17,32);(1,3)(2,5)(4,17,20,30)(6,16)(7,15)(8,26,29,18)(9,24)(10,12)(11,22)(13,19,31,23)(14,25)(21,28,32,27);(1,4)(2,8)(3,13)(5,21)(6,23)(7,20)(9,28)(10,27)(11,29)(12,18)(14,17)(15,31)(16,30)(19,25)(22,32)(24,26);(1,5)(2,3)(4,18)(6,24)(7,22)(8,17)(9,16)(10,14)(11,15)(12,25)(13,28)(19,32)(20,26)(21,23)(27,31)(29,30)Polycyclic presentation:
2^9|a^2=e,b^2=egh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319831, #72319935, #72319954, … |
| maximal normal subgroup of | #72319831, #72319935, #72319954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378699);