Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,12)(7,8)(9,20)(10,24)(13,28)(14,31)(15,32)(16,17)(18,22)(19,23)(21,27)(25,34)(26,29)(30,35)(33,36);(5,6)(7,12)(8,19)(9,23)(10,26)(11,13)(14,20)(15,24)(16,31)(17,32)(18,28)(21,35)(22,33)(25,36)(27,29)(30,34);(2,4)(6,13)(12,28)(14,30)(15,26)(19,33)(23,36)(29,32)(31,35);(5,7)(8,9)(10,27)(11,18)(12,19)(14,23)(15,26)(16,20)(17,24)(21,34)(22,25)(28,33)(29,35)(30,36)(31,32)Polycyclic presentation:
2^9|e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,e^d=eg,g^d=gh,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72234321, #72234374, D4×#2263, … |
| maximal normal subgroup of | #72234321, #72234374, D4×#2263, … |
| maximal quotient of | #72322880, #72322886 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420369);