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 | |
8877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,21)(16,18)(17,20)(19,22)(23,27)(24,25)(26,28);(1,2)(3,4)(6,9)(10,11)(13,14,18,16)(15,19,22,21)(17,24,25,20)(23,27,28,26);(5,6)(7,10)(8,9)(11,12)(14,16)(15,22)(20,24)(23,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28);(5,7)(6,11)(8,12)(9,10)(13,16,18,14)(15,21,22,19)(17,20,25,24)(23,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=gi,b^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72279656, #72302940, #72302966, … |
| maximal normal subgroup of | #72279656, #72302940, #72302966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875454);