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 | |
4618 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)(13,14)(15,17)(16,19)(18,20)(22,25)(23,27);(6,9)(10,11)(13,14,17,15)(16,19,20,18)(23,27)(26,28);(6,9)(7,12)(14,15)(16,20)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,15,17,14)(16,18,20,19)(21,22)(23,26)(24,25)(27,28);(5,6)(7,11)(8,9)(10,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=fg,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 | #72300531, #72300542, #72300559, … |
| maximal normal subgroup of | #72300531, #72300542, #72300559, … |
| maximal quotient of | #72329969, #72330233, #72331281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4041158);