Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,22)(16,20)(17,21)(18,27)(19,26)(23,24)(25,28);(2,4)(5,6,8,9)(7,10,11,12)(13,14,17,21)(15,23,25,19)(16,27,18,20)(22,26,28,24);(6,9)(7,11)(13,15,18,26,17,25,16,24)(14,19,20,22,21,23,27,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,27)(17,21)(18,20)(22,26)(23,25)(24,28);(6,9)(7,11)(13,16,17,18)(14,20,21,27)(15,24,25,26)(19,22,23,28)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263923, #72263982, #72264011 |
| maximal normal subgroup of | #72263923, #72263982, #72264011 |
| maximal quotient of | #72264930 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513618);