Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418498 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15304 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11)(3,13)(4,20,23,9)(5,22,26,24)(6,10,14,12)(8,18)(15,30)(16,27)(17,29)(19,28)(21,25)(31,32);(1,3,14,17,7,19,6,18)(2,8,12,13,11,29,10,28)(4,21,24,15,23,31,22,27)(5,25,9,30,26,32,20,16);(1,4)(2,9)(3,15)(5,10)(6,22)(7,23)(8,25)(11,20)(12,26)(13,30)(14,24)(16,28)(17,31)(18,21)(19,27)(29,32);(1,5,6,20,7,26,14,9)(2,4,10,22,11,23,12,24)(3,16,17,25,19,30,18,32)(8,27,13,21,29,15,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318806, #72318864, #72318894, … |
| maximal normal subgroup of | #72318806, #72318864, #72318894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329499);