Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422927 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15310 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1915 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)(3,7)(4,13)(5,14)(6,12)(8,11)(9,18)(10,21)(15,19)(16,29)(17,24)(20,26)(22,25)(23,31)(27,30)(28,32);(1,3,7,2,8,12,6,11)(4,15,18,9,19,13,17,24)(5,10,21,14,22,26,20,25)(16,23,32,28,31,29,30,27);(1,4,6,18,8,19,7,17)(2,9,3,13,11,24,12,15)(5,16,21,30,22,31,20,32)(10,23,14,27,26,29,25,28);(1,5)(2,10)(3,14)(4,16)(6,21)(7,20)(8,22)(9,23)(11,26)(12,25)(13,27)(15,28)(17,32)(18,30)(19,31)(24,29)Polycyclic presentation:
2^9|b^2=ei,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215912, #72215974, #72215995, … |
| maximal normal subgroup of | #72215912, #72215974, #72215995, … |
| maximal quotient of | C2×#422928, #72252139, #72265227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330124);