Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427936 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15376 |
| 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 | |
1300 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)(3,8)(4,16,18,28)(5,20)(6,10)(7,11)(9,19,27,21)(12,25)(13,30,17,32)(14,24)(15,22)(23,31,26,29);(1,3)(2,8)(4,17)(5,12)(6,15)(7,14)(9,26)(10,22)(11,24)(13,18)(16,30)(19,31)(20,25)(21,29)(23,27)(28,32);(3,12)(5,6)(8,22)(10,20)(13,29)(14,15)(16,28)(17,31)(19,21)(23,30)(24,25)(26,32);(1,4)(2,9)(3,13,14,17)(5,19)(6,21)(7,18)(8,23,24,26)(10,28)(11,27)(12,29,15,31)(16,20)(22,30,25,32)Polycyclic presentation:
2^9|a^2=e,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,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 | #72237064, #72237065, #72237090, … |
| maximal normal subgroup of | #72237064, #72237065, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335132);