Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14718 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2198 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)(3,10)(4,17)(5,12)(6,11)(7,13)(8,15)(9,20)(14,30)(16,24)(18,27)(19,29)(21,28)(22,25)(23,31)(26,32);(1,3)(2,8)(4,18)(5,22)(6,15)(7,16)(9,26)(10,12)(11,24)(13,25)(14,19)(17,23)(20,31)(21,32)(27,29)(28,30);(1,4,5,19,7,21,6,20)(2,9,11,28,13,29,12,17)(3,14,15,18,16,31,22,32)(8,23,10,27,25,30,24,26);(1,3)(2,10)(4,14)(5,15)(6,22)(7,16)(8,12)(9,27)(11,25)(13,24)(17,23)(18,19)(20,32)(21,31)(26,29)(28,30)Polycyclic presentation:
2^9|c^2=e,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^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240435, #72240442, #72240445, … |
| maximal normal subgroup of | #72240435, #72240442, #72240445, … |
| maximal quotient of | #72251307, #72255647, #72256026 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301826);