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 | |
7652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(13,14,18,19)(15,20,23,27)(16,21,24,28)(17,25,26,22);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(16,24)(20,27)(22,25);(6,9)(10,12)(14,19)(15,23)(21,28)(22,25);(7,11)(10,12)(13,15)(14,20)(16,22)(17,21)(18,23)(19,27)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,22)(17,20)(18,24)(19,28)(23,25)(26,27);(5,6)(7,10)(8,9)(11,12)(13,17)(14,22)(15,21)(16,20)(18,26)(19,25)(23,28)(24,27)Polycyclic presentation:
2^9|a^2=h,b^2=gh,e^2=i,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238198, #72238215, #72238218, … |
| maximal normal subgroup of | #72238198, #72238215, #72238218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7911214);