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 | |
8108 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)(15,21)(16,26)(17,22)(18,20)(19,23)(24,25)(27,28);(1,2,3,4)(13,15)(14,19)(16,24)(17,25)(18,23)(20,21)(22,27)(26,28);(6,9)(10,11)(14,20)(15,23)(17,26)(24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,27)(17,25)(18,23)(19,20)(22,24)(26,28);(5,7,8,12)(6,10,9,11)(13,16,18,22)(14,17,20,26)(15,24,23,27)(19,25,21,28);(13,17)(14,22)(15,25)(16,20)(18,26)(19,27)(21,24)(23,28)Polycyclic presentation:
2^9|b^2=gi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72326289, #72327271, #72330752 |
| maximal normal subgroup of | #72326289, #72327271, #72330752 |
| maximal quotient of | #72326292, #72326612, #72326727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7673687);