Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13867 |
| prime radical | #63090 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,14,6,12,7,13)(3,15,19,29,20,9,18,30)(4,21,23,32,22,10,27,17)(5,24,25,16,26,11,28,31);(1,3)(2,9)(4,11)(5,17)(6,20)(7,19)(8,18)(10,25)(12,15)(13,30)(14,29)(16,23)(21,28)(22,24)(26,32)(27,31);(1,4)(2,10)(3,16)(5,9)(6,22)(7,27)(8,23)(11,19)(12,21)(13,32)(14,17)(15,26)(18,24)(20,31)(25,30)(28,29);(1,5)(2,11)(3,17)(4,9)(6,26)(7,25)(8,28)(10,19)(12,24)(13,31)(14,16)(15,22)(18,21)(20,32)(23,29)(27,30)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215938, #72219682, #72219688, … |
| maximal normal subgroup of | #72215938, #72219682, #72219688, … |
| maximal quotient of | C2×#359548, #72343075 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266744);