Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418858 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2009 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)(3,11,13,7)(4,15,17,8)(5,10)(6,9)(12,29)(14,27,19,23)(16,32)(18,28,20,25)(21,26)(22,30)(24,31);(1,3)(2,7)(4,16)(5,19)(6,13)(8,24)(9,11)(10,27)(12,18)(14,21)(15,32)(17,31)(20,30)(22,25)(23,26)(28,29);(1,4)(2,8)(3,12)(5,20)(6,17)(7,22)(9,15)(10,28)(11,29)(13,30)(14,31)(16,19)(18,21)(23,32)(24,27)(25,26);(5,21)(7,11)(10,26)(14,19)(16,31)(18,20)(22,29)(25,28)Polycyclic presentation:
2^9|a^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285055, #72285845, #72286231, … |
| maximal normal subgroup of | #72285055, #72285845, #72286231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326054);