Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 |
| prime radical | C2×#3192 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2090 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)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,28)(23,26,30,34)(24,27,32,29)(31,33,36,35);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,26)(24,31)(25,30)(27,33)(28,34)(29,35)(32,36);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(8,16)(11,20)(13,15)(17,19)(22,28)(23,30)(27,29)(31,36);(7,14)(10,18)(13,15)(17,19)(22,28)(26,34)(27,29)(33,35)Polycyclic presentation:
2^9|a^2=hi,b^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282257, #72282320, #72282404, … |
| maximal normal subgroup of | #72282257, #72282320, #72282404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415988);