Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420424 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2699 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15299 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,4)(5,17)(7,11,20,26)(8,13)(9,23)(12,28,29,32)(14,16)(15,30,24,31)(18,27)(19,25)(21,22);(1,3)(2,8,10,4)(5,12,18,29)(6,13)(7,21)(9,15)(11,16,26,22)(14,20)(17,32)(19,31,23,30)(24,25)(27,28);(1,4,6,8)(2,3)(5,15,18,24)(7,22,20,16)(9,12)(10,13)(11,21)(14,26)(17,31)(19,32,23,28)(25,29)(27,30);(1,5)(2,9)(3,12)(4,15)(6,18)(7,23)(8,24)(10,25)(11,27)(13,29)(14,31)(16,32)(17,26)(19,20)(21,30)(22,28)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319962, #72319976, #72320059, … |
| maximal normal subgroup of | #72319962, #72319976, #72320059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329113);