Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4328 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)(7,8)(9,10)(11,12)(13,14,17,18)(15,19,20,16)(21,22,25,28)(23,26,24,27);(2,4)(6,10)(8,11)(14,18)(16,19)(21,23,25,24)(22,26,28,27);(1,2)(3,4)(13,15)(14,16)(17,20)(18,19)(21,22)(23,26)(24,27)(25,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,26)(24,27)(25,28);(5,8,9,11)(6,7,10,12)(13,16)(14,15)(17,19)(18,20)(21,24,25,23)(22,27,28,26)Polycyclic presentation:
2^9|a^2=g,b^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232308, #72232339, #72232346, … |
| maximal normal subgroup of | #72232308, #72232339, #72232346, … |
| maximal quotient of | #72328367, #72328771, #72328774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 563854);