Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,27)(24,26)(25,28)(29,32)(30,35)(31,33)(34,36);(5,7,9,14)(6,10,12,18)(8,15,16,13)(11,19,20,17)(22,26)(23,30)(24,32)(27,34)(28,29)(33,35);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,23,25,30)(22,27,28,35)(24,31,32,36)(26,33,29,34);(6,12)(10,18)(11,20)(17,19)(22,28)(26,29)(27,35)(33,34);(23,30)(27,35)(31,36)(33,34)Polycyclic presentation:
2^9|b^2=hi,c^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72304713, #72304728, #72304793, … |
| maximal normal subgroup of | #72304713, #72304728, #72304793, … |
| maximal quotient of | #72305070 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341880);