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 | |
7910 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)(5,6,9,10)(7,8,11,12)(13,14,18,20)(15,19,22,21)(16,25,27,17)(23,24,26,28);(1,2,3,4)(13,15)(14,19)(16,26)(17,24)(18,22)(20,21)(23,27)(25,28);(6,10)(7,11)(14,20)(15,22)(17,25)(23,26);(5,7)(6,8)(9,11)(10,12)(13,15)(14,21)(16,23)(17,24)(18,22)(19,20)(25,28)(26,27);(5,8)(6,7)(9,12)(10,11)(13,16)(14,17)(15,23)(18,27)(19,28)(20,25)(21,24)(22,26);(13,17)(14,16)(15,24)(18,25)(19,26)(20,27)(21,23)(22,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72297211, #72330752 |
| maximal normal subgroup of | #72297211, #72330752 |
| maximal quotient of | #72297229, #72297502, #72297956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7654077);