Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14911 |
| 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 | |
2225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(5,6)(10,22)(11,18)(14,29)(15,19)(16,20)(21,24)(23,28)(25,26)(27,30);(1,2)(3,13,15,9)(4,10)(5,18)(6,8)(7,11)(12,21,19,24)(14,28,30,23)(16,25)(17,26)(20,22)(27,31,29,32);(1,3)(2,9)(4,14)(5,12)(6,19)(7,15)(8,21)(10,23)(11,13)(16,27)(17,30)(18,24)(20,29)(22,31)(25,32)(26,28);(1,4)(2,10)(3,14)(5,16)(6,20)(7,17)(8,22)(9,23)(11,26)(12,27)(13,28)(15,30)(18,25)(19,29)(21,31)(24,32)Polycyclic presentation:
2^9|b^2=h,e^2=i,f^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,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237692, #72240437, #72240449, … |
| maximal normal subgroup of | #72237692, #72240437, #72240449, … |
| maximal quotient of | #72237699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312484);