Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(11,15)(13,16)(17,18,23,27)(19,24,28,32)(20,21,25,30)(22,26,31,29);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(17,19)(18,24)(20,22)(21,26)(23,28)(25,31)(27,32)(29,30);(1,2,5,3)(4,6,8,7)(9,11)(10,13)(12,15)(14,16)(19,28)(22,31)(24,32)(26,29);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21)(18,25)(19,29)(20,27)(22,24)(23,30)(26,28)(31,32);(17,22)(18,26)(19,20)(21,24)(23,31)(25,28)(27,29)(30,32)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72312512, #72326661, #72342333, … |
| maximal normal subgroup of | #72312512, #72326661, #72342333, … |
| maximal quotient of | #72326686, #72326750, #72326980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8255806);