Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500378 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3109 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,26,28)(23,30,32,36)(24,29,25,34)(27,33,35,31);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,31,26,32,24,33)(22,27,29,30,28,35,34,36);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(22,28)(27,35)(29,34)(30,36);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,24,26,25)(22,29,28,34)(23,31,32,33)(27,36,35,30);(7,13)(10,16)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72314752, #72314776, #72314788, … |
| maximal normal subgroup of | #72314752, #72314776, #72314788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434042);