Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2380 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2380 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13,16,8)(4,17,21,26)(5,11)(6,10)(7,12)(9,20,27,19)(14,25)(15,28,22,24)(18,23)(29,31)(30,32);(1,3)(2,8)(4,18)(5,15)(6,22)(7,16)(9,25)(10,28)(11,24)(12,13)(14,20)(17,29)(19,30)(21,31)(23,26)(27,32);(1,4,6,20,7,21,5,19)(2,9,11,26,12,27,10,17)(3,14,15,18,16,30,22,31)(8,23,28,25,13,29,24,32);(2,10,12,11)(8,24,13,28)(9,17,27,26)(23,32,29,25);(1,5,7,6)(2,11,12,10)(3,15,16,22)(4,19,21,20)(8,24,13,28)(9,26,27,17)(14,18,30,31)(23,32,29,25)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,c^b=cfg,d^b=df,g^b=gi,h^b=hi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279307, #72279391, #72279606, … |
| maximal normal subgroup of | #72279307, #72279391, #72279606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314830);