Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470941 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16719 |
| 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 | |
731 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)(3,11)(4,15,18,8)(5,19)(6,10)(7,13)(9,20)(12,26,31,32)(14,27)(16,24,17,25)(21,29,23,30)(22,28);(1,3)(2,7,10,22)(4,12,18,31)(5,13)(6,14)(8,23)(9,11,19,27)(15,21)(16,29,17,30)(20,28)(24,26)(25,32);(1,4,5,17,6,18,20,16)(2,8,19,24,10,15,9,25)(3,12,28,29,14,31,13,30)(7,21,11,26,22,23,27,32);(4,16)(5,20)(8,24)(9,19)(11,27)(12,29)(13,28)(15,25)(17,18)(21,32)(23,26)(30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251689, #72251695, #72251701, … |
| maximal normal subgroup of | #72251689, #72251695, #72251701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378137);