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 | |
7524 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)(15,17)(16,21)(18,20)(19,23)(22,25)(24,27)(26,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,20)(18,23)(21,27)(24,26)(25,28);(6,9)(10,12)(14,20)(17,19)(21,28)(25,27);(7,11)(10,12)(13,15)(14,19)(16,24)(17,20)(18,23)(21,25)(22,26)(27,28);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,22)(17,25)(18,26)(19,27)(20,28)(23,24);(5,6)(7,10)(8,9)(11,12)(13,17,18,19)(14,15,20,23)(16,25,26,27)(21,22,28,24)Polycyclic presentation:
2^9|b^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72236865, #72238144, #72238189, … |
| maximal normal subgroup of | #72236865, #72238144, #72238189, … |
| maximal quotient of | #72263405, #72263479 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7557527);