Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301891 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1132 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)(3,11)(4,7)(5,15)(6,10)(8,13)(9,12)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(1,3,12,14,6,13,5,4)(2,7,15,8,10,16,9,11)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,30,31,32);(3,13)(4,14)(7,16)(8,11)(17,20)(18,23)(19,26,27,28)(21,29)(22,30,25,32)(24,31);(1,4,5,13,6,14,12,3)(2,8,9,7,10,11,15,16)(18,24)(22,25)(23,31)(30,32)Polycyclic presentation:
2^9|b^2=eh,c^2=f,d^2=ei,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,d^c=dh,g^c=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72268335, #72268351, #72268376, … |
| maximal normal subgroup of | #72268335, #72268351, #72268376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209103);