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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8740 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)(15,22)(16,19)(17,20)(18,21)(23,28)(24,27)(25,26);(1,2)(3,4)(13,14)(15,22)(16,19)(17,25)(18,21)(20,26)(23,27)(24,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,16)(14,19,21,22)(17,23,26,24)(20,27,25,28);(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,19,21,22)(17,24,26,23)(20,27,25,28);(13,14)(15,19)(16,22)(17,20)(18,21)(23,27)(24,28)(25,26);(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)Polycyclic presentation:
2^9|c^2=i,d^2=h,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 | #72256612, #72256688, #72256690, … |
| maximal normal subgroup of | #72256612, #72256688, #72256690, … |
| maximal quotient of | #72263402, #72263537, #72323329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7539163);