Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462060 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16415 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,11,6,12)(3,8,15,24,14,23,19,22)(4,9,17,28,18,27,20,26)(13,29,16,25,30,21,31,32);(1,3)(2,8,11,23)(4,16)(5,19)(6,15)(7,14)(9,25,27,32)(10,22,12,24)(13,17)(18,31)(20,30)(21,28,29,26);(1,4)(2,9)(3,13,14,30)(5,17)(6,20)(7,18)(8,21,23,29)(10,28)(11,27)(12,26)(15,16,19,31)(22,25,24,32);(2,10)(5,6)(8,22)(9,26)(11,12)(13,30)(15,19)(17,20)(21,25)(23,24)(27,28)(29,32)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251784, #72251789, #72251794, … |
| maximal normal subgroup of | #72251784, #72251789, #72251794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369256);