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 | |
7900 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)(6,8)(10,11)(13,14,18,20)(15,16,22,24)(17,19,26,25)(21,23,28,27);(1,2,3,4)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24)(25,27)(26,28);(5,6,9,8)(7,11,12,10)(13,16,18,24)(14,15,20,22)(17,23,26,27)(19,21,25,28);(5,7,9,12)(6,10,8,11)(13,16)(14,15)(17,23)(18,24)(19,21)(20,22)(25,28)(26,27);(5,8,9,6)(7,11,12,10)(13,14)(15,16)(17,25)(18,20)(19,26)(21,27)(22,24)(23,28);(13,17,18,26)(14,19,20,25)(15,21,22,28)(16,23,24,27)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=hi,e^2=hi,f^2=i,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 | #72236858, #72238167, #72238219, … |
| maximal normal subgroup of | #72236858, #72238167, #72238219, … |
| maximal quotient of | #72281037, #72281335, #72297138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7804878);