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 | |
7685 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)(13,14)(15,16)(17,25)(18,19)(20,26)(21,23)(22,28)(24,27);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,19)(16,23)(20,25)(24,28);(5,6,8,10)(7,9,11,12)(14,19)(15,21)(20,25)(22,27);(7,11)(9,12)(13,15)(14,16)(17,22)(18,21)(19,23)(20,24)(25,28)(26,27);(5,7,8,11)(6,9,10,12)(13,16)(14,15)(17,24)(18,23)(19,21)(20,22)(25,27)(26,28);(5,6)(7,9)(8,10)(11,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,25)(21,27)(23,28)Polycyclic presentation:
2^9|b^2=gh,c^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72263085, #72297126 |
| maximal normal subgroup of | #72263085, #72297126 |
| maximal quotient of | #72263103, #72263126, #72297156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7919406);