Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484139 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,14,10,15,24,27,11,16,8,21,23,32,22,12)(7,17,18,26,19,31,35,29,20,33,25,34,36,13,28,30);(6,12,16,27)(7,18,20,25)(8,22)(9,24)(10,23)(14,15,21,32)(17,33)(19,28,36,35)(26,30)(29,34);(1,2,3,4);(5,7,9,25,10,19,24,28,11,20,8,18,23,36,22,35)(6,13,21,30,15,17,12,26,16,31,14,29,32,33,27,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=ei,e^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=de,e^b=eh,g^b=gh,h^b=hi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286580, #72286784, D4×#2180, … |
| maximal normal subgroup of | #72286580, #72286784, D4×#2180, … |
| maximal quotient of | #72287192, #72287441, #72309487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405775);