Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419211 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15268 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1910 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)(3,12,13,23)(4,16)(5,8,21,10)(7,22,25,15)(9,20)(14,27)(17,24)(18,26)(19,28)(29,30)(31,32);(1,3,5,15,6,13,21,22)(2,7,10,12,11,25,8,23)(4,17,18,30,20,31,19,14)(9,27,26,32,16,29,28,24);(2,8)(3,13)(5,21)(7,23)(9,26)(10,11)(12,25)(14,30)(16,28)(18,19)(24,27)(29,32);(1,4)(2,9)(3,14)(5,19)(6,20)(7,24)(8,26)(10,28)(11,16)(12,29)(13,30)(15,31)(17,22)(18,21)(23,27)(25,32)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,e^d=ei,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72219488, #72219493, … |
| maximal normal subgroup of | #72213913, #72219488, #72219493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326407);