Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#120 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,4,13)(5,15,16,9)(7,14,8,12)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(1,3,6,4)(2,7,10,8)(5,14,16,12)(9,13,15,11)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(22,28)(29,32);(1,4,6,3)(2,8,10,7)(5,12,16,14)(9,11,15,13)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,e^d=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72268398, #72268401, #72268402, … |
| maximal normal subgroup of | #72268398, #72268401, #72268402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 155683);