Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #34980 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,9)(10,12)(13,14,18,23)(15,19,25,26)(16,21,27,20)(17,22,28,24);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,20)(18,25)(21,28)(23,26)(24,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,24)(17,26)(18,27)(19,28)(21,23)(22,25);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,22)(17,26)(18,27)(19,28)(20,23)(24,25);(13,15)(14,19)(16,24)(17,21)(18,25)(20,28)(22,27)(23,26);(13,17)(14,22)(15,21)(16,26)(18,28)(19,27)(20,25)(23,24)Polycyclic presentation:
2^9|a^2=i,d^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72302359, #72302940, #72302952, … |
| maximal normal subgroup of | #72302359, #72302940, #72302952, … |
| maximal quotient of | #72326446, #72327134, #72328048 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538783);