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 | |
8610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,22)(18,20)(19,21)(23,24)(25,27)(26,28);(1,2)(3,4)(13,15)(14,19)(16,20)(17,25)(18,21)(22,24)(23,26)(27,28);(6,9)(10,11)(14,20)(15,21)(17,26)(24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,23)(18,21)(19,20)(22,24)(25,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,21)(15,20)(17,24)(18,19)(22,25)(23,28)(26,27);(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,28)(21,25)Polycyclic presentation:
2^9|a^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72261985, #72262081, #72262185, … |
| maximal normal subgroup of | #72261985, #72262081, #72262185, … |
| maximal quotient of | #72297888, #72297972, #72298088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7723066);