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 | |
7830 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)(13,14,18,19)(15,16,21,24)(17,20,26,27)(22,23,28,25);(1,2,3,4)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,27)(26,28);(6,9)(10,11)(14,19)(15,21)(20,27)(22,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,25)(18,24)(19,21)(20,28)(22,27)(23,26);(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,28)(24,25)Polycyclic presentation:
2^9|a^2=i,b^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72238195, #72238200, #72238207, … |
| maximal normal subgroup of | #72238195, #72238200, #72238207, … |
| maximal quotient of | #72327896, #72329140, #72329584 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7803697);