Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479788 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17248 |
| 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 | |
645 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)(3,13,15,24)(4,17)(5,10)(6,11)(7,12)(8,16,25,22)(9,20)(14,23,30,32)(18,29,31,26)(19,28)(21,27);(1,3)(2,8)(4,18,21,31)(5,22)(6,16)(7,15)(9,26,28,29)(10,24)(11,13)(12,25)(14,19,30,20)(17,23,27,32);(1,4,5,20,7,21,6,19)(2,9,11,17,12,28,10,27)(3,14,16,18,15,30,22,31)(8,23,24,26,25,32,13,29);(3,15)(4,19)(5,6)(9,17)(10,11)(13,24)(14,18)(20,21)(23,26)(27,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,h^c=hi,e^d=ei,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251686, #72251693, #72251705, … |
| maximal normal subgroup of | #72251686, #72251693, #72251705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386984);