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 | |
9038 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)(13,14,18,20)(15,19,23,26)(16,21,24,27)(17,22,25,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,18,23)(14,19,20,26)(16,17,24,25)(21,22,27,28);(6,9)(10,11)(14,20)(15,23)(16,24)(22,28);(13,16)(14,21)(15,17)(18,24)(19,22)(20,27)(23,25)(26,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,26)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,17)(14,22)(15,16)(18,25)(19,21)(20,28)(23,24)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72267576, #72267581, #72267583, … |
| maximal normal subgroup of | #72267576, #72267581, #72267583, … |
| maximal quotient of | #72327865, #72329104, #72329563 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7629107);