Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3892 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)(5,6,8,9)(7,11,12,10)(14,15)(18,19)(21,22)(23,25)(24,27)(26,28);(6,9)(10,11)(13,14,17,15)(16,18,20,19)(21,22,25,23)(24,27,28,26);(6,9)(7,12)(16,20)(18,19)(22,23)(24,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,15,17,14)(16,18,20,19)(21,23,25,22)(24,26,28,27);(5,6)(7,11)(8,9)(10,12)(13,16)(14,18)(15,19)(17,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72326667, #72326717, #72326744, … |
| maximal normal subgroup of | #72326667, #72326717, #72326744, … |
| maximal quotient of | #72328765, #72330583 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4041123);