Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 |
| prime radical | C2×#2408 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4128 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)(14,17)(15,19)(21,22,25,24)(23,27,28,26);(1,2,3,4)(6,9)(7,12)(13,14,16,17)(15,20,19,18)(21,22)(23,26)(24,25)(27,28);(1,2,3,4)(7,12)(10,11)(15,19)(18,20)(22,24)(26,27);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,17)(18,19)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,15,16,19)(14,18,17,20)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72300478, #72300521, #72300525, … |
| maximal normal subgroup of | #72300478, #72300521, #72300525, … |
| maximal quotient of | #72328724, #72331393, #72332678 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3814393);