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 | |
7585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,9)(10,11)(14,19)(16,23)(21,27)(24,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,16,20,23)(17,21,26,27)(22,24,25,28);(5,6,8,9)(7,11,12,10)(14,19)(16,23)(21,27)(24,28);(7,12)(10,11)(13,15)(14,16)(17,25)(18,20)(19,23)(21,28)(22,26)(24,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,19)(17,24)(18,23)(21,25)(22,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,17)(14,21)(15,22)(16,24)(18,26)(19,27)(20,25)(23,28)Polycyclic presentation:
2^9|b^2=hi,c^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72312502, #72312537, #72312785, … |
| maximal normal subgroup of | #72312502, #72312537, #72312785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7980329);