Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,10,9)(7,12,8,11)(13,14,16,20,17,21,26,28)(15,22,25,27,24,18,23,19);(1,2,4,3)(5,7,10,8)(6,11,9,12)(13,15)(14,18)(16,25)(17,24)(19,20)(21,22)(23,26)(27,28);(1,3,4,2)(5,6,10,9)(7,12,8,11)(13,15)(14,19)(16,23)(17,24)(18,20)(21,27)(22,28)(25,26);(5,8,10,7)(6,11,9,12)(13,16,17,26)(14,20,21,28)(15,23,24,25)(18,27,22,19);(5,9,10,6)(7,12,8,11)(14,21)(18,22)(19,27)(20,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72307349, #72307411, #72307500, … |
| maximal normal subgroup of | #72307349, #72307411, #72307500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461903);