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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2678 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,18)(17,19)(23,27)(24,28);(2,4)(5,6,8,10)(7,11,12,9)(13,15,17,16)(14,18,19,20)(21,22,25,26)(23,28,27,24);(1,2,3,4)(7,12)(9,11)(14,19)(18,20)(21,23)(22,24)(25,27)(26,28);(5,6)(7,11)(8,10)(9,12)(13,16,17,15)(14,20,19,18)(21,22)(23,24)(25,26)(27,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,20)(16,18)(17,19)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^9|b^2=fgi,c^2=ghi,d^2=i,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 | #72326686, #72326720, #72326727, … |
| maximal normal subgroup of | #72326686, #72326720, #72326727, … |
| maximal quotient of | #72328733 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3937384);