Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394032 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3918 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)(4,10)(5,22)(6,11)(7,12)(8,16)(9,20)(14,30)(15,29)(17,25)(18,24)(19,28)(21,27)(23,31)(26,32);(1,3)(2,8)(4,15)(5,17)(6,16)(7,18)(9,26)(10,23)(11,24)(12,25)(13,22)(14,19)(20,31)(21,32)(27,30)(28,29);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,23)(10,22)(11,27)(12,28)(13,29)(15,16)(17,32)(18,31)(24,26)(25,30);(1,4)(2,10)(3,15)(5,20)(6,19)(7,21)(8,23)(9,22)(11,28)(12,27)(13,26)(14,16)(17,31)(18,32)(24,29)(25,30)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72241161, #72241171, #72241303, … |
| maximal normal subgroup of | #72241161, #72241171, #72241303, … |
| maximal quotient of | C2×#394032, #72316843, #72316940 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301228);