Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428079 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15377 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1168 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)(3,7)(4,9)(5,18)(6,14)(8,11)(10,20)(12,22)(13,21)(15,23)(16,24)(17,25)(19,26)(27,29)(28,31)(30,32);(1,3,7,2,8,14,6,11)(4,9,22,24,17,25,16,12)(5,10,21,13,20,18,19,26)(15,28,31,23,30,27,29,32);(1,4)(2,9)(3,12)(5,15)(6,16)(7,22)(8,17)(10,23)(11,25)(13,27)(14,24)(18,32)(19,29)(20,30)(21,31)(26,28);(1,5,6,21,8,20,7,19)(2,10,3,13,11,18,14,26)(4,15,16,31,17,30,22,29)(9,23,12,27,25,32,24,28)Polycyclic presentation:
2^9|b^2=ei,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338618, #72338624, #72338629 |
| maximal normal subgroup of | #72338618, #72338624, #72338629 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335276);