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 | |
8002 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)(5,6)(7,11)(8,9)(10,12)(13,14,19,23)(15,24,16,20)(17,21,27,26)(18,22,28,25);(1,2,3,4)(13,15,19,16)(14,20,23,24)(17,22,27,25)(18,26,28,21);(6,9)(10,11)(15,16)(17,27)(20,24)(21,26);(5,6)(7,10)(8,9)(11,12)(13,16,19,15)(14,20,23,24)(17,25,27,22)(18,26,28,21);(5,7,8,12)(6,10,9,11)(13,17)(14,21)(15,22)(16,25)(18,20)(19,27)(23,26)(24,28);(13,18)(14,22)(15,21)(16,26)(17,20)(19,28)(23,25)(24,27)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72327640 |
| maximal normal subgroup of | #72327640 |
| maximal quotient of | #72327846, #72328060, #72328248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7643373);