Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #379827 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14255 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2117 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)(3,11,15,7)(4,8)(5,10)(6,9)(12,21,14,19)(13,25,27,20)(16,22)(17,23)(18,24)(26,31,28,30)(29,32);(1,3)(2,7)(4,13)(5,12)(6,15)(8,20)(9,11)(10,19)(14,18)(16,27)(17,26)(21,24)(22,25)(23,30)(28,29)(31,32);(3,12)(5,18)(7,19)(8,22)(10,24)(11,21)(13,26)(14,15)(17,29)(20,31)(25,30)(27,28);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,22)(10,23)(11,25)(12,26)(14,28)(15,27)(18,29)(19,30)(21,31)(24,32)Polycyclic presentation:
2^9|a^2=h,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240445, #72240447, #72240449, … |
| maximal normal subgroup of | #72240445, #72240447, #72240449, … |
| maximal quotient of | #72255660, #72255831 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287023);