Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418772 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,12,6,8)(3,9,13,24,16,25,15,21)(4,17,20,26,19,22,18,10)(14,28,27,23,30,31,29,32);(2,8)(3,13)(4,18)(5,6)(10,26)(11,12)(15,16)(19,20)(21,24)(23,28)(27,29)(31,32);(1,3)(2,9)(4,14)(5,13)(6,15)(7,16)(8,21)(10,23)(11,24)(12,25)(17,31)(18,29)(19,30)(20,27)(22,28)(26,32);(1,4)(2,10)(3,14)(5,20)(6,18)(7,19)(8,22)(9,23)(11,17)(12,26)(13,27)(15,29)(16,30)(21,28)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,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 | #72237052, #72237054, #72237065, … |
| maximal normal subgroup of | #72237052, #72237054, #72237065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325968);