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 | |
6186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32);(3,7)(6,8)(10,14)(11,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(19,26)(22,23)(27,32)(30,31);(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)Polycyclic presentation:
2^9|b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72249501, #72249510, #72249549, … |
| maximal normal subgroup of | #72249501, #72249510, #72249549, … |
| maximal quotient of | #72326587, #72326701, #72327448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9073629);