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 | |
6132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,20)(19,23,25,27)(21,24,29,28)(26,30,31,32);(3,7)(5,8)(10,14)(11,16)(18,20)(23,27)(24,28)(30,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(19,25)(21,29)(23,27)(24,28);(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249434, #72249551, #72256344 |
| maximal normal subgroup of | #72219217, #72219285, #72249434, … |
| maximal quotient of | #72219215, #72262617, #72262793 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837870);