Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,24)(7,12,19,27,17,26,18,33)(8,20,22,28,23,34,21,13)(16,29,32,35,30,36,31,25);(1,2)(3,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,25)(14,27)(15,26)(20,29)(21,31)(22,32)(23,30)(24,33)(28,35)(34,36);(5,8,9,21,11,23,10,22)(6,13,14,20,15,28,24,34)(7,16,18,31,17,30,19,32)(12,25,27,29,26,35,33,36);(2,4);(7,17)(12,26)(16,30)(18,19)(25,35)(27,33)(29,36)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239487, #72250588, #72250592, … |
| maximal normal subgroup of | #72239487, #72250588, #72250592, … |
| maximal quotient of | #72318419, #72320158, C2×#6380098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287294);