Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,16)(15,19,20,18)(22,26)(23,27);(6,9)(10,11)(13,15,17,20)(14,18,16,19)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,23,25,27)(22,24,26,28);(5,7,8,12)(6,10,9,11)(13,16,17,14)(15,19,20,18)(21,22)(23,24)(25,26)(27,28);(5,6)(7,11)(8,9)(10,12)(13,15)(14,19)(16,18)(17,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=fh,d^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72300583, #72300584, #72300619, … |
| maximal normal subgroup of | #72300583, #72300584, #72300619, … |
| maximal quotient of | #72333133 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 701405);