Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418531 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1708 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)(4,18)(5,8)(6,11)(7,12)(9,16)(10,20)(14,23)(15,24)(17,22)(19,26)(21,25)(27,28)(29,32)(30,31);(2,8)(4,19)(5,6)(9,22)(11,12)(13,24)(14,28)(15,16)(18,25)(20,21)(23,32)(29,30);(1,3)(2,9)(4,14)(5,15)(6,16)(7,17)(8,13)(10,23)(11,22)(12,24)(18,27)(19,28)(20,30)(21,29)(25,31)(26,32);(1,4,5,19,7,21,6,20)(2,10,11,25,12,26,8,18)(3,14,15,28,17,29,16,30)(9,23,22,31,24,32,13,27)Polycyclic presentation:
2^9|d^2=e,e^2=i,f^2=i,h^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,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216047, #72216077, … |
| maximal normal subgroup of | #72215936, #72216047, #72216077, … |
| maximal quotient of | #72239632, #72239645, C2×#418531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325727);