Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,16,19)(15,18,20,17)(22,23)(24,28);(6,9)(10,11)(13,15)(14,17)(16,20)(18,19)(22,23)(26,27);(7,12)(10,11)(15,20)(17,18)(21,22,25,23)(24,26,28,27);(1,2,3,4)(5,7,8,12)(6,10,9,11)(13,15)(14,18)(16,20)(17,19)(21,23,25,22)(24,27,28,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|a^2=gh,c^2=g,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
| maximal subgroup of | #72332011, #72332024, #72332057 |
| maximal normal subgroup of | #72332011, #72332024, #72332057 |
| maximal quotient of | #72333153 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4406972);