Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #404013 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14885 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2650 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)(3,13)(4,17)(5,12)(6,10)(7,11)(8,15)(9,20)(14,28)(16,26)(18,25)(19,24)(21,22)(23,30)(27,29)(31,32);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,23)(10,26)(11,22)(12,13)(17,27)(18,31)(19,30)(20,29)(24,32)(25,28);(5,6)(8,22)(10,12)(15,21)(17,24)(18,20)(23,28)(29,31);(1,4)(2,9,11,25)(3,14)(5,18)(6,20)(7,19)(8,23,22,28)(10,24,12,17)(13,27,26,32)(15,29)(16,30)(21,31)Polycyclic presentation:
2^9|d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72234100, #72251414, #72251427, … |
| maximal normal subgroup of | #72234100, #72251414, #72251427, … |
| maximal quotient of | #72234086, #72234098, #72255873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311209);