Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #430345 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15418 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,9)(3,13,16,8,14,24,22,25)(4,17,20,26,21,10,19,28)(15,27,18,23,29,30,31,32);(1,3,6,16,7,14,5,22)(2,8,11,24,12,25,9,13)(4,18,20,29,21,31,19,15)(10,27,28,23,17,30,26,32);(2,9)(3,14)(5,6)(8,13)(10,26)(11,12)(15,29)(17,28)(19,20)(23,27)(24,25)(30,32);(1,4)(2,10,12,17)(3,15,14,29)(5,19)(6,20)(7,21)(8,23)(9,26,11,28)(13,27)(16,18,22,31)(24,30)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251689, #72251691, #72251708, … |
| maximal normal subgroup of | #72251689, #72251691, #72251708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337541);