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 | |
5815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,4)(3,7,8,6)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,29)(30,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,21)(20,28)(22,26)(23,25)(24,32)(27,30)(29,31);(9,11,13,12)(10,14,15,16)(18,23)(19,26)(24,31)(27,28);(2,4)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,30)(31,32);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,14,15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72300328, #72326379, #72334246 |
| maximal normal subgroup of | #72300328, #72326379, #72334246 |
| maximal quotient of | #72300423, #72326431, #72326719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10062994);