Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405282 |
| 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×#3027 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(5,17)(8,19)(9,20)(10,18)(11,23)(12,25)(13,27)(15,28)(21,31)(22,29)(24,30);(1,2)(3,11,14,8)(4,9)(5,18)(6,10)(7,17)(12,24,25,21)(13,19,27,23)(15,29)(16,22)(20,28)(26,30,32,31);(1,3)(2,8)(4,12)(5,13)(6,14)(7,19)(9,21)(10,11)(15,26)(16,25)(17,27)(18,23)(20,30)(22,24)(28,32)(29,31);(1,4)(2,9)(3,12)(5,15)(6,16)(7,20)(8,21)(10,22)(11,24)(13,26)(14,25)(17,28)(18,29)(19,30)(23,31)(27,32)Polycyclic presentation:
2^9|b^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240442, #72240443, #72240449, … |
| maximal normal subgroup of | #72240442, #72240443, #72240449, … |
| maximal quotient of | #72255979 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312478);