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 | |
7532 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)(13,14)(15,16)(17,21)(18,19)(20,24)(22,23)(25,26)(27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,16,20,24)(17,25,26,21)(22,28,27,23);(6,8)(10,12)(14,19)(15,20)(17,26)(23,28);(7,11)(10,12)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,28)(26,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,20)(15,19)(17,23)(18,24)(21,27)(22,25)(26,28);(5,8,9,6)(7,12,11,10)(13,17,18,26)(14,21,19,25)(15,22,20,27)(16,23,24,28)Polycyclic presentation:
2^9|a^2=gh,b^2=h,f^2=i,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 | #72279530, #72297211, #72308580, … |
| maximal normal subgroup of | #72279530, #72297211, #72308580, … |
| maximal quotient of | #72297257, #72297543, #72297977 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7744352);